1 00:00:00,000 --> 00:00:03,190 PROFESSOR TODA: And Calc II. 2 00:00:03,190 --> 00:00:08,189 And I will go ahead and solve some problems today out 3 00:00:08,189 --> 00:00:10,780 of chapter 10 as a review. 4 00:00:10,780 --> 00:00:14,494 5 00:00:14,494 --> 00:00:15,478 Meaning what? 6 00:00:15,478 --> 00:00:22,540 Meaning, that you have section 10.1 followed by 10.2 7 00:00:22,540 --> 00:00:24,740 followed by 10.4. 8 00:00:24,740 --> 00:00:26,920 These ones are required sections, 9 00:00:26,920 --> 00:00:34,520 but I'm putting the material all together as a compact set. 10 00:00:34,520 --> 00:00:38,816 So, if we cannot officially cut between, as I told you, 11 00:00:38,816 --> 00:00:41,720 cut between the sections. 12 00:00:41,720 --> 00:00:47,270 One thing that I did not work examples on, 13 00:00:47,270 --> 00:00:50,400 trusting that you'd remember it was integration. 14 00:00:50,400 --> 00:00:53,270 In particular, I didn't cover integration 15 00:00:53,270 --> 00:00:56,220 of vector valued functions and examples that 16 00:00:56,220 --> 00:00:58,070 are very very important. 17 00:00:58,070 --> 00:01:02,820 Now, do you need to learn something special for that? 18 00:01:02,820 --> 00:01:03,320 No. 19 00:01:03,320 --> 00:01:07,820 But just like you cannot learn organic chemistry without 20 00:01:07,820 --> 00:01:11,540 knowing inorganic chemistry, then you could not know how 21 00:01:11,540 --> 00:01:17,032 to integrate a vector value function r prime of d to get r 22 00:01:17,032 --> 00:01:21,220 of d unless you know calculus one and caluculus two, right? 23 00:01:21,220 --> 00:01:25,680 So let's say first a bunch of formulas 24 00:01:25,680 --> 00:01:30,570 that you use going back to last week's knowledge 25 00:01:30,570 --> 00:01:32,100 what have we learned? 26 00:01:32,100 --> 00:01:38,900 We work with regular curves in r3. 27 00:01:38,900 --> 00:01:42,200 And in particular if they are part of R2, 28 00:01:42,200 --> 00:01:45,170 they are plain curves. 29 00:01:45,170 --> 00:01:47,690 I want to encourage you to ask questions 30 00:01:47,690 --> 00:01:50,090 about the example [INAUDIBLE] now. 31 00:01:50,090 --> 00:01:57,110 In the review session we have applications [INAUDIBLE] 32 00:01:57,110 --> 00:01:58,870 from 2 2 3. 33 00:01:58,870 --> 00:02:00,870 What was a regular curve? 34 00:02:00,870 --> 00:02:04,185 Is anybody willing to tell me what a regular curve was? 35 00:02:04,185 --> 00:02:08,092 Was it vector value function? 36 00:02:08,092 --> 00:02:09,699 Do you like big r or little r? 37 00:02:09,699 --> 00:02:10,699 STUDENT: Doesn't matter. 38 00:02:10,699 --> 00:02:11,980 PROFESSOR TODA: Big r of t. 39 00:02:11,980 --> 00:02:14,120 Vector value function. 40 00:02:14,120 --> 00:02:17,940 x of t [INAUDIBLE] You know, I told you that sometimes we 41 00:02:17,940 --> 00:02:19,290 use brackets here. 42 00:02:19,290 --> 00:02:25,040 Sometimes we use round parentheses depending 43 00:02:25,040 --> 00:02:31,130 how you represent a vector in r3 in our book they use brackets, 44 00:02:31,130 --> 00:02:37,070 but in other calculus books, they use round parentheses 45 00:02:37,070 --> 00:02:38,510 around it. 46 00:02:38,510 --> 00:02:44,270 So these are the coordinates of the moving particle in time. 47 00:02:44,270 --> 00:02:47,600 Doesn't have to be a specific object, could be a fly, 48 00:02:47,600 --> 00:02:50,780 could be just a particle, anything 49 00:02:50,780 --> 00:02:58,000 in physical motion between this point a of b equals a and b 50 00:02:58,000 --> 00:03:00,140 of t equals b. 51 00:03:00,140 --> 00:03:02,336 So at time a and time b you are there. 52 00:03:02,336 --> 00:03:03,210 What have we learned? 53 00:03:03,210 --> 00:03:11,280 We've learned that a regular curve means its differentiable 54 00:03:11,280 --> 00:03:15,120 and the derivative is continuous, it's a c1 function. 55 00:03:15,120 --> 00:03:16,290 And what else? 56 00:03:16,290 --> 00:03:19,934 The derivative of the position vector 57 00:03:19,934 --> 00:03:22,850 called velocity never vanishes. 58 00:03:22,850 --> 00:03:26,670 So it's different from 0 for every t in the interval 59 00:03:26,670 --> 00:03:30,050 that you take, like ab. 60 00:03:30,050 --> 00:03:31,840 That's a regular curve. 61 00:03:31,840 --> 00:03:38,720 Regular curve was something we talked about at least 5 times. 62 00:03:38,720 --> 00:03:44,170 The point is how do we see the backwards process? 63 00:03:44,170 --> 00:03:52,160 That means if somebody gives you the velocity of a vector curve, 64 00:03:52,160 --> 00:03:55,186 they ask you for the position vector. 65 00:03:55,186 --> 00:03:57,320 So let's see an example. 66 00:03:57,320 --> 00:04:02,990 Integration example 1 says I gave you 67 00:04:02,990 --> 00:04:07,820 the veclocity vector or a certain law of motion 68 00:04:07,820 --> 00:04:09,030 that I don't know. 69 00:04:09,030 --> 00:04:13,175 I just know the velocity vector is being 1 over 1 70 00:04:13,175 --> 00:04:15,600 plus t squared. 71 00:04:15,600 --> 00:04:17,055 Should I put the brace here? 72 00:04:17,055 --> 00:04:19,000 An angular bracket? 73 00:04:19,000 --> 00:04:20,720 One over one plus t squared. 74 00:04:20,720 --> 00:04:32,863 And I'm gonna put a cosign on 2t, and t squared 75 00:04:32,863 --> 00:04:37,500 plus equal to minus t. 76 00:04:37,500 --> 00:04:43,290 And somebody says, that's all I know for P 77 00:04:43,290 --> 00:04:47,210 on an arbitrary real integral. 78 00:04:47,210 --> 00:04:54,980 And we know via the 0 as being even. 79 00:04:54,980 --> 00:05:03,091 Let's say it's even as 0 0 and that 80 00:05:03,091 --> 00:05:06,610 takes a little bit of thinking. 81 00:05:06,610 --> 00:05:09,000 I don't know. 82 00:05:09,000 --> 00:05:20,260 How about a 1, which would be just k. 83 00:05:20,260 --> 00:05:24,450 Using this velocity vector find me being normal, 84 00:05:24,450 --> 00:05:26,750 which means find the position vector 85 00:05:26,750 --> 00:05:29,690 corresponding to this velocity. 86 00:05:29,690 --> 00:05:31,370 What is this? 87 00:05:31,370 --> 00:05:34,340 It's actually initial value 88 00:05:34,340 --> 00:05:40,620 STUDENT: [INAUDIBLE] 1, 1, and 1? 89 00:05:40,620 --> 00:05:42,400 PROFESSOR TODA: 0, what is it? 90 00:05:42,400 --> 00:05:44,061 When place 0 in? 91 00:05:44,061 --> 00:05:45,432 STUDENT: Yeah. 92 00:05:45,432 --> 00:05:47,670 [INTERPOSING VOICES] 93 00:05:47,670 --> 00:05:49,530 STUDENT: Are these the initial conditions 94 00:05:49,530 --> 00:05:50,802 for the location, or-- 95 00:05:50,802 --> 00:05:51,885 PROFESSOR TODA: I'm sorry. 96 00:05:51,885 --> 00:05:58,780 I wrote r the intial condition for the location. 97 00:05:58,780 --> 00:06:01,040 Thank you so much, OK? 98 00:06:01,040 --> 00:06:04,520 I probably would've realized it as soon as possible. 99 00:06:04,520 --> 00:06:07,030 Not the initial velocity I wanted to give you, 100 00:06:07,030 --> 00:06:11,120 but the initial position. 101 00:06:11,120 --> 00:06:17,730 All right, so how do I get to the r of d? 102 00:06:17,730 --> 00:06:20,000 I would say integrate, and when I integrate, 103 00:06:20,000 --> 00:06:25,680 I have to keep in mind that I have to add the constants. 104 00:06:25,680 --> 00:06:26,320 Right? 105 00:06:26,320 --> 00:06:27,000 OK. 106 00:06:27,000 --> 00:06:29,442 So from v, v is our priority. 107 00:06:29,442 --> 00:06:32,400 108 00:06:32,400 --> 00:06:37,876 It follows that r will be-- who tells me? 109 00:06:37,876 --> 00:06:42,514 Do you guys remember the integral of 1 plus t squared? 110 00:06:42,514 --> 00:06:43,462 STUDENT: [INAUDIBLE] 111 00:06:43,462 --> 00:06:45,358 PROFESSOR TODA: So that's the inverse. 112 00:06:45,358 --> 00:06:49,012 Or, I'll write it [? arc tan, ?] and I'm very happy that you 113 00:06:49,012 --> 00:06:51,220 remember that, but there are many students who don't. 114 00:06:51,220 --> 00:06:54,530 If you feel you don't, that means that you have to open 115 00:06:54,530 --> 00:06:59,740 the -- where? -- Between chapters 5 and chapter 7. 116 00:06:59,740 --> 00:07:03,680 You have all these integration chapters-- 117 00:07:03,680 --> 00:07:05,974 the main ones over there. 118 00:07:05,974 --> 00:07:08,140 It's a function definted on the whole real interval, 119 00:07:08,140 --> 00:07:11,860 so I don't care to worry about it. 120 00:07:11,860 --> 00:07:14,700 This what we call an IVP, initial value problem. 121 00:07:14,700 --> 00:07:18,650 122 00:07:18,650 --> 00:07:20,945 So what kind of problem is that? 123 00:07:20,945 --> 00:07:23,030 It's a problem like somebody would 124 00:07:23,030 --> 00:07:29,570 give you knowing that f prime of t is the little f, 125 00:07:29,570 --> 00:07:32,820 and knowing that big f of 0 is the initial value 126 00:07:32,820 --> 00:07:37,340 for your function of find f. 127 00:07:37,340 --> 00:07:42,860 So you have actually an initial value problem of the calc 128 00:07:42,860 --> 00:07:47,160 that you've seen in previous class. 129 00:07:47,160 --> 00:07:54,680 arctangent of t plus c1 and then if you miss the c1 in general, 130 00:07:54,680 --> 00:07:59,690 this can mess up the whole thing because-- see, in your case, 131 00:07:59,690 --> 00:08:02,280 you're really lucky. 132 00:08:02,280 --> 00:08:06,650 If you plug in the 0 here, what are you gonna have? 133 00:08:06,650 --> 00:08:10,500 You're gonna have arctangent of 0, and that is 0. 134 00:08:10,500 --> 00:08:12,732 So in that case c1 is just 0. 135 00:08:12,732 --> 00:08:15,190 And [? three ?] [? not ?] and if you forgot it would not be 136 00:08:15,190 --> 00:08:19,520 the end of the world, but if you forgot it in general, 137 00:08:19,520 --> 00:08:20,870 it would be a big problem. 138 00:08:20,870 --> 00:08:22,990 So don't forget about the constant. 139 00:08:22,990 --> 00:08:25,260 When you integrate-- the familiar of antiderivatives 140 00:08:25,260 --> 00:08:26,450 is cosine 2t. 141 00:08:26,450 --> 00:08:29,980 142 00:08:29,980 --> 00:08:32,510 I know you know it. 143 00:08:32,510 --> 00:08:35,870 1/2 sine of t. 144 00:08:35,870 --> 00:08:37,240 Am I done? 145 00:08:37,240 --> 00:08:40,390 No, I should say plus C2. 146 00:08:40,390 --> 00:08:43,039 And finally the familiar of antiderivatives of t 147 00:08:43,039 --> 00:08:45,700 squared plus e to minus t. 148 00:08:45,700 --> 00:08:48,230 STUDENT: 2t minus e to the negative t. 149 00:08:48,230 --> 00:08:50,120 PROFESSOR TODA: No, integral of. 150 00:08:50,120 --> 00:08:52,570 So what's the integral of-- 151 00:08:52,570 --> 00:08:53,700 STUDENT: t 2 squared. 152 00:08:53,700 --> 00:08:57,860 PROFESSOR TODA: t cubed over 3-- minus, excellent. 153 00:08:57,860 --> 00:09:01,310 Now, do you want one of you guys almost 154 00:09:01,310 --> 00:09:03,427 kill me during the weekend. 155 00:09:03,427 --> 00:09:04,010 But that's OK. 156 00:09:04,010 --> 00:09:06,340 I mean, this problem had something 157 00:09:06,340 --> 00:09:08,100 to do with integral minus. 158 00:09:08,100 --> 00:09:12,130 He put that integral of e to the minus t was equal to minus t. 159 00:09:12,130 --> 00:09:14,450 So pay attention to the sign. 160 00:09:14,450 --> 00:09:16,925 Remember that integral of e to the at, 161 00:09:16,925 --> 00:09:22,220 the t is to the at over a plus. 162 00:09:22,220 --> 00:09:23,020 Right? 163 00:09:23,020 --> 00:09:26,810 OK, so this is what you have, a minus plus C3. 164 00:09:26,810 --> 00:09:28,770 Pay attention also to the exam. 165 00:09:28,770 --> 00:09:30,708 Because in the exams, when you rush, 166 00:09:30,708 --> 00:09:33,030 you make lots of mistakes like that. 167 00:09:33,030 --> 00:09:36,810 R of 0 is even. 168 00:09:36,810 --> 00:09:43,090 So the initial position is given as C1. 169 00:09:43,090 --> 00:09:44,820 I'm replacing in my formula. 170 00:09:44,820 --> 00:09:49,430 It's going to be C1, C2, and what? 171 00:09:49,430 --> 00:09:52,264 When I replace the 0 here, what am I going to get? 172 00:09:52,264 --> 00:09:53,930 STUDENT: You're going to get negative 1. 173 00:09:53,930 --> 00:09:59,060 PROFESSOR TODA: Minus 1 plus C3. 174 00:09:59,060 --> 00:10:03,040 Note that I fabricated this example, so that C3 is not 175 00:10:03,040 --> 00:10:04,440 going to be 0. 176 00:10:04,440 --> 00:10:06,790 I wanted some customs to be zero and some customs 177 00:10:06,790 --> 00:10:10,235 to not be 0, just for you to realize it's 178 00:10:10,235 --> 00:10:12,560 important to pay attention. 179 00:10:12,560 --> 00:10:14,720 OK, minus 1 plus C3. 180 00:10:14,720 --> 00:10:22,490 And then I have 0, 0, 1 as given as initial position. 181 00:10:22,490 --> 00:10:28,130 So what do you get by solving this linear system that's 182 00:10:28,130 --> 00:10:29,230 very simple? 183 00:10:29,230 --> 00:10:32,300 In general, you can get more complicated stuff. 184 00:10:32,300 --> 00:10:35,740 C1 is 0, C2 is 0, C3 is a-- 185 00:10:35,740 --> 00:10:36,360 STUDENT: 2. 186 00:10:36,360 --> 00:10:37,110 PROFESSOR TODA: 2. 187 00:10:37,110 --> 00:10:38,860 And so it was a piece of cake. 188 00:10:38,860 --> 00:10:40,880 What is my formula? 189 00:10:40,880 --> 00:10:43,590 If you leave it like that, generally you're 190 00:10:43,590 --> 00:10:44,670 going to get full credit. 191 00:10:44,670 --> 00:10:47,400 What would you need to do to get full credit? 192 00:10:47,400 --> 00:10:53,424 STUDENT: Rt is equal to R10 plus 1/2 sine of 2t plus tq-- 193 00:10:53,424 --> 00:10:55,424 PROFESSOR TODA: Precisely, and thank you so much 194 00:10:55,424 --> 00:10:56,810 for your help. 195 00:10:56,810 --> 00:11:01,550 So you have R10 of t, 1/2 sine of 2t 196 00:11:01,550 --> 00:11:09,905 and t cubed over 3 minus e to the minus e plus 2. 197 00:11:09,905 --> 00:11:11,650 And close, and that's it. 198 00:11:11,650 --> 00:11:14,240 And box your answer. 199 00:11:14,240 --> 00:11:16,260 So I got the long motion back. 200 00:11:16,260 --> 00:11:22,000 Similarly, you could find, if somebody gives you 201 00:11:22,000 --> 00:11:27,840 the acceleration of a long motion and asks you 202 00:11:27,840 --> 00:11:29,830 this is the acceleration. 203 00:11:29,830 --> 00:11:31,910 And I give you some initial values. 204 00:11:31,910 --> 00:11:34,520 And you have to find first the velocity, 205 00:11:34,520 --> 00:11:36,270 going backwards one step. 206 00:11:36,270 --> 00:11:39,680 And from the velocity, backwards a second step, 207 00:11:39,680 --> 00:11:42,050 get the position vector. 208 00:11:42,050 --> 00:11:44,050 And that sounds a little bit more elaborate. 209 00:11:44,050 --> 00:11:47,110 But it doesn't have to be a long computation. 210 00:11:47,110 --> 00:11:50,960 In general, we do not focus on giving you 211 00:11:50,960 --> 00:11:54,100 an awfully long computation. 212 00:11:54,100 --> 00:11:58,870 We just want to test your understanding of the concepts. 213 00:11:58,870 --> 00:12:04,490 And having this in mind, I picked another example. 214 00:12:04,490 --> 00:12:08,570 I would like to see what that is. 215 00:12:08,570 --> 00:12:14,377 And the initial velocity will be given in this case. 216 00:12:14,377 --> 00:12:16,812 This is what I was thinking a little bit ahead of that. 217 00:12:16,812 --> 00:12:22,890 So somebody gives you the acceleration in the velocity 218 00:12:22,890 --> 00:12:31,130 vector at 0 and is asking you to find the velocity vector So 219 00:12:31,130 --> 00:12:36,300 let me give it to you for t between 0 and 2 pi. 220 00:12:36,300 --> 00:12:38,050 I give you the acceleration vector, 221 00:12:38,050 --> 00:12:40,370 it will be nice and sassy. 222 00:12:40,370 --> 00:12:45,690 Let's see, that's going to be cosine of t, sine of t and 0. 223 00:12:45,690 --> 00:12:48,340 And you'll say, oh, I know how to do those. 224 00:12:48,340 --> 00:12:49,770 Of course you know. 225 00:12:49,770 --> 00:12:52,370 But I want you to pay attention to the constraints 226 00:12:52,370 --> 00:12:53,140 of integration. 227 00:12:53,140 --> 00:12:58,250 This is why I do this kind of exercise again. 228 00:12:58,250 --> 00:13:02,810 So what do we have for V of t. 229 00:13:02,810 --> 00:13:09,980 V of 0 is-- somebody will say, let's give something nice, 230 00:13:09,980 --> 00:13:15,530 and let's say this would be-- I have no idea what I want. 231 00:13:15,530 --> 00:13:21,770 Let's say i, j, and that's it. 232 00:13:21,770 --> 00:13:24,590 233 00:13:24,590 --> 00:13:26,950 How do you do that? 234 00:13:26,950 --> 00:13:27,640 V of t. 235 00:13:27,640 --> 00:13:30,006 Let's integrate together. 236 00:13:30,006 --> 00:13:31,530 You don't like this? 237 00:13:31,530 --> 00:13:35,270 I hope that by now, you've got used to it. 238 00:13:35,270 --> 00:13:38,560 A bracket, I'm doing a bracket, like in the book. 239 00:13:38,560 --> 00:13:42,700 So sine t plus a constant. 240 00:13:42,700 --> 00:13:45,090 What's the integral of sine, class? 241 00:13:45,090 --> 00:13:48,200 V equals sine t plus a constant. 242 00:13:48,200 --> 00:13:51,380 And C3 is a constant. 243 00:13:51,380 --> 00:13:52,790 And there I go. 244 00:13:52,790 --> 00:13:54,670 You say, oh my god, what am I having? 245 00:13:54,670 --> 00:13:58,360 V of 0-- is as a vector, I presented it 246 00:13:58,360 --> 00:14:03,810 in the canonical standard basis as 1, 1, and 0. 247 00:14:03,810 --> 00:14:07,350 So from that one, you can jump to this one 248 00:14:07,350 --> 00:14:10,740 and say, yes, I'm going to plug in 0, see what I get. 249 00:14:10,740 --> 00:14:13,376 In the general formula, when you plug in 0, 250 00:14:13,376 --> 00:14:19,070 you get C1-- what is cosine of 0? 251 00:14:19,070 --> 00:14:22,470 Minus 1, I have here, plus C2. 252 00:14:22,470 --> 00:14:27,900 And C3, that is always there. 253 00:14:27,900 --> 00:14:34,530 And then V of 0 is what I got here. 254 00:14:34,530 --> 00:14:39,730 V of 0 has to be compared to what your initial data was. 255 00:14:39,730 --> 00:14:48,380 So C1 is 1, C2 is 2, and C3 is-- 256 00:14:48,380 --> 00:14:51,460 So let me replace it. 257 00:14:51,460 --> 00:15:05,427 I say the answer will be-- cosine t plus 1, sine t plus 2, 258 00:15:05,427 --> 00:15:13,315 and the constants. 259 00:15:13,315 --> 00:15:19,240 260 00:15:19,240 --> 00:15:24,740 But then somebody, who is really an experimental guy, 261 00:15:24,740 --> 00:15:25,539 says well-- 262 00:15:25,539 --> 00:15:26,830 STUDENT: You have it backwards. 263 00:15:26,830 --> 00:15:28,971 It's sine of t plus 1, and then you 264 00:15:28,971 --> 00:15:31,054 have the cosine of t plus 2. 265 00:15:31,054 --> 00:15:32,095 PROFESSOR TODA: Oh, yeah. 266 00:15:32,095 --> 00:15:35,470 267 00:15:35,470 --> 00:15:36,490 Wait a minute. 268 00:15:36,490 --> 00:15:41,550 This is-- I miscopied looking up. 269 00:15:41,550 --> 00:15:52,208 So I have sine t, I was supposed to-- minus cosine t 270 00:15:52,208 --> 00:15:56,550 and I'm done. 271 00:15:56,550 --> 00:15:58,380 So thank you for telling me. 272 00:15:58,380 --> 00:16:04,030 So sum t plus 1 minus cosine t plus 2 and 0 273 00:16:04,030 --> 00:16:13,070 are the functions that I put here by replacing C1, C2, C3. 274 00:16:13,070 --> 00:16:15,260 And then, somebody says, wait a minute, 275 00:16:15,260 --> 00:16:18,500 now let me give you V of 0. 276 00:16:18,500 --> 00:16:20,860 Let me give you R of 0. 277 00:16:20,860 --> 00:16:22,495 We have zeroes already there. 278 00:16:22,495 --> 00:16:25,290 279 00:16:25,290 --> 00:16:28,800 And you were supposed to get R from here. 280 00:16:28,800 --> 00:16:36,320 So what is R of t, the position vector, find it. 281 00:16:36,320 --> 00:16:38,000 V of t is given. 282 00:16:38,000 --> 00:16:40,230 Actually, it's given by you, because you found it 283 00:16:40,230 --> 00:16:41,940 at the previous step. 284 00:16:41,940 --> 00:16:46,300 And R of 0 is given as well. 285 00:16:46,300 --> 00:16:59,218 And let's say that would be-- let's say 1, 1, and 1. 286 00:16:59,218 --> 00:17:02,630 287 00:17:02,630 --> 00:17:05,220 So what do you need to do next? 288 00:17:05,220 --> 00:17:14,390 289 00:17:14,390 --> 00:17:18,069 You have R prime given. 290 00:17:18,069 --> 00:17:22,140 That leaves you to integrate to get R t. 291 00:17:22,140 --> 00:17:24,520 And R of t is going to be what? 292 00:17:24,520 --> 00:17:29,030 Who is going to tell me what I have to write down? 293 00:17:29,030 --> 00:17:39,475 Minus cosine t plus t plus-- let's use the constant K1 294 00:17:39,475 --> 00:17:40,948 integration. 295 00:17:40,948 --> 00:17:42,421 And then what? 296 00:17:42,421 --> 00:17:43,410 STUDENT: Sine of t. 297 00:17:43,410 --> 00:17:45,380 PROFESSOR TODA: I think it's minus sine, right? 298 00:17:45,380 --> 00:17:56,120 Minus sine of t plus 2t plus K2 and K3, right? 299 00:17:56,120 --> 00:18:04,200 So R of 0 is going to be what? 300 00:18:04,200 --> 00:18:07,930 First of all, we use this piece of information. 301 00:18:07,930 --> 00:18:12,480 Second of all, we identify from the formula we got. 302 00:18:12,480 --> 00:18:16,305 So from the formula I got, just plugging in 0, 303 00:18:16,305 --> 00:18:22,690 it should come out straight as minus 1 plus K1. 304 00:18:22,690 --> 00:18:28,070 0 for this guy, 0 for the second term, K2 and K3. 305 00:18:28,070 --> 00:18:31,830 306 00:18:31,830 --> 00:18:36,510 So who is helping me solve the system really quickly? 307 00:18:36,510 --> 00:18:39,750 K1 is 2. 308 00:18:39,750 --> 00:18:41,030 K2 is-- 309 00:18:41,030 --> 00:18:41,720 STUDENT: 1. 310 00:18:41,720 --> 00:18:44,010 PROFESSOR TODA: K3 is 1. 311 00:18:44,010 --> 00:18:50,610 And I'm going back to R and replace it. 312 00:18:50,610 --> 00:18:55,470 And that's my final answer for this two-step problem. 313 00:18:55,470 --> 00:18:57,870 So I have a two-step integration from the acceleration 314 00:18:57,870 --> 00:19:00,270 to the velocity, from the velocity 315 00:19:00,270 --> 00:19:05,080 to the position vector. 316 00:19:05,080 --> 00:19:08,250 Minus cosine t plus t plus 2. 317 00:19:08,250 --> 00:19:11,630 Remind me, because I have a tendency to miscopy, 318 00:19:11,630 --> 00:19:13,280 an I looking in the right place? 319 00:19:13,280 --> 00:19:14,340 Yes. 320 00:19:14,340 --> 00:19:24,930 So I have minus sine t plus 2t plus 1 and K3 is one. 321 00:19:24,930 --> 00:19:29,751 So this is the process you are supposed to remember 322 00:19:29,751 --> 00:19:32,160 for the rest of the semester. 323 00:19:32,160 --> 00:19:33,420 It's not a hard one. 324 00:19:33,420 --> 00:19:36,960 It's something that everybody should master. 325 00:19:36,960 --> 00:19:38,060 Is it hard? 326 00:19:38,060 --> 00:19:39,810 How many of you understood this? 327 00:19:39,810 --> 00:19:41,820 Please raise hands. 328 00:19:41,820 --> 00:19:45,110 Oh, no problem, good. 329 00:19:45,110 --> 00:19:52,160 Now would you tell me-- I'm not going to ask you 330 00:19:52,160 --> 00:19:53,800 what kind of motion this is. 331 00:19:53,800 --> 00:19:57,030 It's a little bit close to a circular motion but not 332 00:19:57,030 --> 00:19:58,376 a circular motion. 333 00:19:58,376 --> 00:20:00,670 However, can you tell me anything interesting 334 00:20:00,670 --> 00:20:04,800 about the type of trajectory that I have, in terms 335 00:20:04,800 --> 00:20:06,380 of the acceleration vector? 336 00:20:06,380 --> 00:20:10,820 The acceleration vector is beautiful, 337 00:20:10,820 --> 00:20:13,840 just like in the case of the washer. 338 00:20:13,840 --> 00:20:18,870 That was a vector that-- like this 339 00:20:18,870 --> 00:20:20,860 would be the circular motion. 340 00:20:20,860 --> 00:20:23,040 The acceleration would be this unique vector 341 00:20:23,040 --> 00:20:25,050 that comes inside. 342 00:20:25,050 --> 00:20:26,920 Is this going outside or coming inside? 343 00:20:26,920 --> 00:20:29,600 Is it a unit vector? 344 00:20:29,600 --> 00:20:32,720 Yes, it is a unit vector. 345 00:20:32,720 --> 00:20:37,430 So suppose that I'm looking at the trajectory, 346 00:20:37,430 --> 00:20:40,290 if it were more or less a motion that has 347 00:20:40,290 --> 00:20:44,760 to do with mixing into a bowl. 348 00:20:44,760 --> 00:20:48,830 Would this go inside or outside? 349 00:20:48,830 --> 00:20:51,960 Towards the outside or towards the inside? 350 00:20:51,960 --> 00:20:57,650 I plugged j-- depends on what I'm looking at, in terms 351 00:20:57,650 --> 00:21:00,150 of surface that I'm on, right? 352 00:21:00,150 --> 00:21:01,560 Do you remember from last time we 353 00:21:01,560 --> 00:21:04,055 had that helix that was on a cylinder. 354 00:21:04,055 --> 00:21:07,920 And we asked ourselves, how is that [INAUDIBLE] pointing? 355 00:21:07,920 --> 00:21:11,780 And it was pointing outside of the cylinder, 356 00:21:11,780 --> 00:21:16,052 in the direction towards the outside. 357 00:21:16,052 --> 00:21:26,930 Coming back to the review, there are 358 00:21:26,930 --> 00:21:31,440 several things I'd like to review but not all of them. 359 00:21:31,440 --> 00:21:34,439 Because some of the examples we have there, 360 00:21:34,439 --> 00:21:38,150 you understood them really well. 361 00:21:38,150 --> 00:21:40,300 I was very proud of you, and I saw 362 00:21:40,300 --> 00:21:43,758 that you finished-- almost all of you 363 00:21:43,758 --> 00:21:45,746 finished the homework number one. 364 00:21:45,746 --> 00:21:49,100 So I was looking outside at homework number 365 00:21:49,100 --> 00:21:53,180 two that is over these three sections. 366 00:21:53,180 --> 00:21:58,471 So I was hoping you would ask me today, between two and three, 367 00:21:58,471 --> 00:22:00,856 if you have any difficulties with homework two. 368 00:22:00,856 --> 00:22:03,730 That's due February 11. 369 00:22:03,730 --> 00:22:12,730 And then the latest homework that I posted yesterday, I 370 00:22:12,730 --> 00:22:14,980 don't know how many of you logged in. 371 00:22:14,980 --> 00:22:18,620 But last night I posted a homework 372 00:22:18,620 --> 00:22:21,800 that is getting a huge extended deadline, which 373 00:22:21,800 --> 00:22:23,370 is the 28th of February. 374 00:22:23,370 --> 00:22:29,010 Because somebody's birthday is February 29. 375 00:22:29,010 --> 00:22:34,740 I was just thinking why would somebody need be a whole month? 376 00:22:34,740 --> 00:22:37,300 You would need the whole month to have a good view 377 00:22:37,300 --> 00:22:39,020 of the whole chapter 11. 378 00:22:39,020 --> 00:22:40,970 I sent you the videos for chapter 11. 379 00:22:40,970 --> 00:22:43,540 And for chapter 11, you have this huge homework 380 00:22:43,540 --> 00:22:46,770 which is 49 problems. 381 00:22:46,770 --> 00:22:50,430 So please do not, do not leave it 382 00:22:50,430 --> 00:22:52,260 to the last five days or six days, 383 00:22:52,260 --> 00:22:55,710 because it's going to kill you. 384 00:22:55,710 --> 00:22:57,495 There are people who say, I can finish 385 00:22:57,495 --> 00:22:58,620 this in the next five days. 386 00:22:58,620 --> 00:23:00,320 I know you can. 387 00:23:00,320 --> 00:23:01,950 I know you can, I don't doubt it. 388 00:23:01,950 --> 00:23:04,420 That's why I left you so much freedom. 389 00:23:04,420 --> 00:23:07,610 But you have-- today is the second or the third? 390 00:23:07,610 --> 00:23:10,860 So practically you have 25 days to work on this. 391 00:23:10,860 --> 00:23:15,200 On the 28th at 11 PM it's going to close. 392 00:23:15,200 --> 00:23:18,820 I would work a few problems every other day. 393 00:23:18,820 --> 00:23:22,050 Because I need a break, so I would alternate. 394 00:23:22,050 --> 00:23:25,030 But don't leave it-- even if you have help, 395 00:23:25,030 --> 00:23:27,690 especially if you have help, like a tutor or tutoring 396 00:23:27,690 --> 00:23:30,280 services here that are free in the department. 397 00:23:30,280 --> 00:23:32,220 Do not leave it to the last days. 398 00:23:32,220 --> 00:23:35,160 Because you're putting pressure on yourself, on your brain, 399 00:23:35,160 --> 00:23:37,221 on your tutor, on everybody. 400 00:23:37,221 --> 00:23:37,720 Yes sir. 401 00:23:37,720 --> 00:23:38,640 STUDENT: So that's homework three? 402 00:23:38,640 --> 00:23:40,223 PROFESSOR TODA: That's homework three, 403 00:23:40,223 --> 00:23:43,305 and it's a huge homework over chapter 11. 404 00:23:43,305 --> 00:23:45,610 STUDENT: You said there are 49 problems? 405 00:23:45,610 --> 00:23:49,157 PROFESSOR TODA: I don't remember exactly but 47, 49. 406 00:23:49,157 --> 00:23:50,240 I don't remember how many. 407 00:23:50,240 --> 00:23:52,850 STUDENT: Between 45 and 50. 408 00:23:52,850 --> 00:23:55,910 PROFESSOR TODA: Between 45 and 50, yes. 409 00:23:55,910 --> 00:23:59,210 If you encounter any bug-- although there shouldn't 410 00:23:59,210 --> 00:24:02,020 be bugs, maybe 1 in 1,000. 411 00:24:02,020 --> 00:24:04,530 If you encounter any bug that the programmer 412 00:24:04,530 --> 00:24:09,710 of those problems may have accidentally put in, 413 00:24:09,710 --> 00:24:11,370 you let me know. 414 00:24:11,370 --> 00:24:13,580 So I can contact them. 415 00:24:13,580 --> 00:24:17,390 If there is a problem that I consider shouldn't be there, 416 00:24:17,390 --> 00:24:19,790 I will eliminate that later on. 417 00:24:19,790 --> 00:24:23,450 But hopefully, everything will be doable, 418 00:24:23,450 --> 00:24:28,096 everything will be fair and you will be able to solve it. 419 00:24:28,096 --> 00:24:32,140 420 00:24:32,140 --> 00:24:34,590 Any questions? 421 00:24:34,590 --> 00:24:37,015 Particular questions from the homework? 422 00:24:37,015 --> 00:24:39,925 423 00:24:39,925 --> 00:24:43,805 STUDENT: [INAUDIBLE] is it to parametrize a circle of a set, 424 00:24:43,805 --> 00:24:47,860 like of a certain radius on the xy-plane? 425 00:24:47,860 --> 00:24:49,260 PROFESSOR TODA: Shall we do that? 426 00:24:49,260 --> 00:24:53,228 Do you want me to do that in general, in xy-plane, OK. 427 00:24:53,228 --> 00:24:55,224 STUDENT: [INAUDIBLE] in the xy-plane. 428 00:24:55,224 --> 00:24:59,220 429 00:24:59,220 --> 00:25:04,660 PROFESSOR TODA: xy-plane and then what was the equation? 430 00:25:04,660 --> 00:25:10,162 Was it like a equals sine of t or a equals sine of bt? 431 00:25:10,162 --> 00:25:12,432 Because it's a little bit different, 432 00:25:12,432 --> 00:25:15,790 depending on how the parametrization was given. 433 00:25:15,790 --> 00:25:17,165 What's your name again, I forgot. 434 00:25:17,165 --> 00:25:18,915 I don't know what to refer you. 435 00:25:18,915 --> 00:25:19,540 STUDENT: Ryder. 436 00:25:19,540 --> 00:25:22,397 437 00:25:22,397 --> 00:25:24,730 PROFESSOR TODA: Was that part of what's due on the 11th? 438 00:25:24,730 --> 00:25:27,890 STUDENT: It doesn't-- yes, it doesn't give a revision set. 439 00:25:27,890 --> 00:25:29,050 It says-- 440 00:25:29,050 --> 00:25:33,000 PROFESSOR TODA: Let me quickly read-- find parametrization 441 00:25:33,000 --> 00:25:38,440 of the circle of radius 7 in the xy-plane, centered at 3, 1, 442 00:25:38,440 --> 00:25:40,615 oriented counterclockwise. 443 00:25:40,615 --> 00:25:43,238 The point 10, 1 should be connected-- 444 00:25:43,238 --> 00:25:44,675 STUDENT: Just one more second. 445 00:25:44,675 --> 00:25:45,633 PROFESSOR TODA: Do you mind if I put it. 446 00:25:45,633 --> 00:25:47,070 I'll take good care of it. 447 00:25:47,070 --> 00:25:48,028 I won't drop it. 448 00:25:48,028 --> 00:25:51,860 449 00:25:51,860 --> 00:25:57,880 So the point-- parametrization of the circle of radius 450 00:25:57,880 --> 00:26:01,980 7 in the xy-plane, centered at 3, 1. 451 00:26:01,980 --> 00:26:12,090 So circle centered at-- and I'll say it x0, 1 0, being 3, 1. 452 00:26:12,090 --> 00:26:16,340 453 00:26:16,340 --> 00:26:19,040 No, because then I'm solving your problem. 454 00:26:19,040 --> 00:26:20,710 But I'm solving your problem anyway, 455 00:26:20,710 --> 00:26:23,480 even if I change change the numbers. 456 00:26:23,480 --> 00:26:26,060 457 00:26:26,060 --> 00:26:27,960 Why don't I change the numbers, and then 458 00:26:27,960 --> 00:26:30,850 you do it for the given numbers. 459 00:26:30,850 --> 00:26:33,880 Let's say 1, 0. 460 00:26:33,880 --> 00:26:39,610 And it's the same type of problem, right? 461 00:26:39,610 --> 00:26:42,662 Oriented counterclockwise. 462 00:26:42,662 --> 00:26:43,370 That's important. 463 00:26:43,370 --> 00:26:51,720 464 00:26:51,720 --> 00:26:54,290 So you have circle radius 7. 465 00:26:54,290 --> 00:26:56,850 I think people could have any other, 466 00:26:56,850 --> 00:27:01,370 because problems are-- sometimes you get a random assignment. 467 00:27:01,370 --> 00:27:05,472 So you have R equals 2, let's say. 468 00:27:05,472 --> 00:27:08,330 469 00:27:08,330 --> 00:27:14,230 And you have the point, how to make up something. 470 00:27:14,230 --> 00:27:21,292 The point corresponding to t equals 471 00:27:21,292 --> 00:27:29,794 0 will be given as you have [INAUDIBLE], 1, 0, whatever. 472 00:27:29,794 --> 00:27:32,000 OK? 473 00:27:32,000 --> 00:27:36,850 Use the t as the parameter for all your answers. 474 00:27:36,850 --> 00:27:39,180 So use t as a parameter for all your answers, 475 00:27:39,180 --> 00:27:42,915 and the answers are written in the interactive field as x of t 476 00:27:42,915 --> 00:27:45,310 equals what and y of t equals what, 477 00:27:45,310 --> 00:27:47,414 and it's waiting for you to fill them in. 478 00:27:47,414 --> 00:27:49,230 You know. 479 00:27:49,230 --> 00:27:54,470 OK, now I was talking to [INAUDIBLE]. 480 00:27:54,470 --> 00:27:56,890 I'm going to give this back to you. 481 00:27:56,890 --> 00:27:57,680 Thank you, Ryan. 482 00:27:57,680 --> 00:28:02,760 So when you said it's a little bit frustrating, 483 00:28:02,760 --> 00:28:07,800 and I agree wit you, that in this variant of webwork 484 00:28:07,800 --> 00:28:10,760 problems you have to enter both of them correctly 485 00:28:10,760 --> 00:28:14,640 in order to say yes, correct. 486 00:28:14,640 --> 00:28:18,070 I was used to another library-- the library was outdated 487 00:28:18,070 --> 00:28:22,480 [INAUDIBLE]-- where if I enter this correctly I get 50% 488 00:28:22,480 --> 00:28:25,665 credit, and if I enter this incorrectly it's not going 489 00:28:25,665 --> 00:28:26,810 to penalize me. 490 00:28:26,810 --> 00:28:29,915 So I a little bit complained about it, 491 00:28:29,915 --> 00:28:32,310 and I was shown the old library where 492 00:28:32,310 --> 00:28:35,550 I can go ahead and go back and assign problems 493 00:28:35,550 --> 00:28:38,870 where you get the answer correct for this one 494 00:28:38,870 --> 00:28:42,130 and incorrect for this one, and you get partial credit. 495 00:28:42,130 --> 00:28:46,590 So I'm probably going to switch to that. 496 00:28:46,590 --> 00:28:47,310 Let's do that. 497 00:28:47,310 --> 00:28:48,530 This is a very good problem. 498 00:28:48,530 --> 00:28:51,755 I'm glad you brought it up. 499 00:28:51,755 --> 00:28:56,850 What have you learned about conics in high school? 500 00:28:56,850 --> 00:28:59,770 You've learned about-- well, it depends. 501 00:28:59,770 --> 00:29:01,380 You've learned about ellipse. 502 00:29:01,380 --> 00:29:03,000 You've learned about hyperbola. 503 00:29:03,000 --> 00:29:04,500 You've learned about parabola. 504 00:29:04,500 --> 00:29:07,190 Some of you put them down for me for extra credit. 505 00:29:07,190 --> 00:29:08,980 I was very happy you did that. 506 00:29:08,980 --> 00:29:10,490 It's a good exercise. 507 00:29:10,490 --> 00:29:12,169 If you have-- Alex, yes? 508 00:29:12,169 --> 00:29:14,210 STUDENT: I was just thinking, does that say 1, 0? 509 00:29:14,210 --> 00:29:17,600 510 00:29:17,600 --> 00:29:19,410 The point corresponding to t0 [INAUDIBLE]? 511 00:29:19,410 --> 00:29:20,450 PROFESSOR TODA: I think that's what I meant. 512 00:29:20,450 --> 00:29:22,060 I don't know, I just came up with it. 513 00:29:22,060 --> 00:29:22,620 I made it. 514 00:29:22,620 --> 00:29:23,227 1, 0. 515 00:29:23,227 --> 00:29:24,310 I make up all my problems. 516 00:29:24,310 --> 00:29:26,120 STUDENT: But the center of the circle isn't 1, 0. 517 00:29:26,120 --> 00:29:27,190 PROFESSOR TODA: Oh, oops. 518 00:29:27,190 --> 00:29:29,742 Yes. 519 00:29:29,742 --> 00:29:31,590 Sorry. 520 00:29:31,590 --> 00:29:33,330 So 2, 0. 521 00:29:33,330 --> 00:29:34,100 No-- 522 00:29:34,100 --> 00:29:35,290 [INTERPOSING VOICES] 523 00:29:35,290 --> 00:29:38,304 PROFESSOR TODA: --because the radius. 524 00:29:38,304 --> 00:29:40,720 This is the problem when you don't think very [INAUDIBLE]. 525 00:29:40,720 --> 00:29:44,240 I always like to make up my own problems. 526 00:29:44,240 --> 00:29:48,400 When an author, when we came up with the problems in the book, 527 00:29:48,400 --> 00:29:51,700 of course we had to think, draw, and make sure they made sense. 528 00:29:51,700 --> 00:29:55,185 But when you just come up with a problem out of the middle 529 00:29:55,185 --> 00:29:57,475 of nowhere-- thank you so much. 530 00:29:57,475 --> 00:29:58,850 Of course, we would have realized 531 00:29:58,850 --> 00:30:01,090 that was nonsense in just a minute. 532 00:30:01,090 --> 00:30:04,470 But it's good that you told me. 533 00:30:04,470 --> 00:30:06,526 So x of t, y of t. 534 00:30:06,526 --> 00:30:10,810 535 00:30:10,810 --> 00:30:11,900 Let's find it. 536 00:30:11,900 --> 00:30:13,353 Based on what? 537 00:30:13,353 --> 00:30:15,560 What is the general equation of a circle? 538 00:30:15,560 --> 00:30:22,340 x minus x0 squared plus y minus y0 squared equals R squared. 539 00:30:22,340 --> 00:30:24,530 And you have learned that in high school. 540 00:30:24,530 --> 00:30:26,590 Am I right or not? 541 00:30:26,590 --> 00:30:27,390 You have. 542 00:30:27,390 --> 00:30:28,240 OK. 543 00:30:28,240 --> 00:30:29,130 Good. 544 00:30:29,130 --> 00:30:36,190 Now, in our case what is x0 and what is y0? 545 00:30:36,190 --> 00:30:40,000 x0 is 1 and y0 is 0. 546 00:30:40,000 --> 00:30:42,860 Because that's why-- I don't know. 547 00:30:42,860 --> 00:30:44,060 I just made it up. 548 00:30:44,060 --> 00:30:47,070 And I said that's the center. 549 00:30:47,070 --> 00:30:49,144 I'll draw. 550 00:30:49,144 --> 00:30:50,810 I should have drawn it in the beginning, 551 00:30:50,810 --> 00:30:54,400 and that would have helped me not come up 552 00:30:54,400 --> 00:31:00,430 with some nonsensical data. 553 00:31:00,430 --> 00:31:01,610 c is 1, 0. 554 00:31:01,610 --> 00:31:03,010 Radius is 2. 555 00:31:03,010 --> 00:31:04,640 So I'm going this way. 556 00:31:04,640 --> 00:31:06,790 What point is this way, guys? 557 00:31:06,790 --> 00:31:08,710 Just by the way. 558 00:31:08,710 --> 00:31:10,240 Because [INAUDIBLE] is 1, 0, right? 559 00:31:10,240 --> 00:31:16,240 And this way the other extreme, the antipode is 3, 0. 560 00:31:16,240 --> 00:31:20,088 So that's exactly what Alexander was saying. 561 00:31:20,088 --> 00:31:22,020 And now it makes sense. 562 00:31:22,020 --> 00:31:25,401 563 00:31:25,401 --> 00:31:26,500 Well, I cannot draw today. 564 00:31:26,500 --> 00:31:27,333 STUDENT: [INAUDIBLE] 565 00:31:27,333 --> 00:31:30,354 566 00:31:30,354 --> 00:31:31,770 PROFESSOR TODA: It looks horrible. 567 00:31:31,770 --> 00:31:37,065 It looks like an egg that is shaped-- disabled egg. 568 00:31:37,065 --> 00:31:41,470 569 00:31:41,470 --> 00:31:42,480 OK. 570 00:31:42,480 --> 00:31:43,060 All right. 571 00:31:43,060 --> 00:31:49,640 So the motion of-- the motion will come like that. 572 00:31:49,640 --> 00:31:53,736 From t equals 0, when I'm here, counterclockwise, 573 00:31:53,736 --> 00:31:57,312 I have to draw-- any kind of circle you have in the homework 574 00:31:57,312 --> 00:32:00,610 should be drawn on the board. 575 00:32:00,610 --> 00:32:06,210 If you have a general, you don't know what the data is. 576 00:32:06,210 --> 00:32:08,800 I want to help you solve the general problem. 577 00:32:08,800 --> 00:32:10,775 For the original problem, which is a circle 578 00:32:10,775 --> 00:32:15,470 of center x, 0, y, 0 and radius R, generic one, 579 00:32:15,470 --> 00:32:19,574 what is the parametrization without data? 580 00:32:19,574 --> 00:32:20,490 Without specific data. 581 00:32:20,490 --> 00:32:23,240 What is the parametrization? 582 00:32:23,240 --> 00:32:25,644 And I want you to pay attention very well. 583 00:32:25,644 --> 00:32:26,930 You are paying attention. 584 00:32:26,930 --> 00:32:29,730 You are very careful today. 585 00:32:29,730 --> 00:32:31,190 [INAUDIBLE] 586 00:32:31,190 --> 00:32:33,940 So what do you have? 587 00:32:33,940 --> 00:32:35,740 STUDENT: Cosine. 588 00:32:35,740 --> 00:32:37,520 PROFESSOR TODA: Before that cosine 589 00:32:37,520 --> 00:32:39,580 there is an R, excellent. 590 00:32:39,580 --> 00:32:43,980 So [INAUDIBLE] there R cosine of t. 591 00:32:43,980 --> 00:32:46,550 I'm not done. 592 00:32:46,550 --> 00:32:47,475 What do I put here? 593 00:32:47,475 --> 00:32:48,342 STUDENT: Over d. 594 00:32:48,342 --> 00:32:49,300 PROFESSOR TODA: No, no. 595 00:32:49,300 --> 00:32:51,276 I'm continuing. 596 00:32:51,276 --> 00:32:52,240 STUDENT: Plus x0. 597 00:32:52,240 --> 00:32:53,690 PROFESSOR TODA: Plus x0. 598 00:32:53,690 --> 00:32:57,460 And R sine t plus y0. 599 00:32:57,460 --> 00:32:59,560 Who taught me that? 600 00:32:59,560 --> 00:33:02,640 First of all, this is not unique. 601 00:33:02,640 --> 00:33:03,650 It's not unique. 602 00:33:03,650 --> 00:33:05,920 I could put sine t here and cosine t here 603 00:33:05,920 --> 00:33:08,650 and it would be the same type of parametrization. 604 00:33:08,650 --> 00:33:11,020 But we usually put the cosine first 605 00:33:11,020 --> 00:33:13,500 because we look at the x-axis corresponding 606 00:33:13,500 --> 00:33:17,766 to the cosine and the y-axis corresponding to the sine. 607 00:33:17,766 --> 00:33:20,845 If I don't know that, because I happen to know that 608 00:33:20,845 --> 00:33:24,170 from when I was 16 in high school, if I don't know that, 609 00:33:24,170 --> 00:33:25,380 what do I know? 610 00:33:25,380 --> 00:33:27,550 I cook up my own parametrization. 611 00:33:27,550 --> 00:33:29,130 And that's a very good thing. 612 00:33:29,130 --> 00:33:31,160 And I'm glad Ryan asked about that. 613 00:33:31,160 --> 00:33:33,360 How does one come up with this? 614 00:33:33,360 --> 00:33:34,360 Do we have to memorize? 615 00:33:34,360 --> 00:33:38,190 In mathematics, thank god, we don't memorize much. 616 00:33:38,190 --> 00:33:41,730 The way we cook up things is just from, in this case, 617 00:33:41,730 --> 00:33:44,930 from the Pythagorean theorem of-- no. 618 00:33:44,930 --> 00:33:47,140 Pythagorean theorem of trigonometry? 619 00:33:47,140 --> 00:33:49,170 The fundamental identity of trigonometry, 620 00:33:49,170 --> 00:33:52,840 which is the same thing as the Pythagorean theorem. 621 00:33:52,840 --> 00:33:55,300 What's the fundamental identity of trigonometry? 622 00:33:55,300 --> 00:33:58,210 Cosine squared plus sin squared equals 1. 623 00:33:58,210 --> 00:34:03,681 If I have a problem like that, I must 624 00:34:03,681 --> 00:34:08,810 have that this is R cosine t and this is R sine t. 625 00:34:08,810 --> 00:34:11,310 Because when I take the red guys and I 626 00:34:11,310 --> 00:34:13,818 square them and I add them together, 627 00:34:13,818 --> 00:34:17,650 I'm going to have R squared. 628 00:34:17,650 --> 00:34:18,940 All righty, good. 629 00:34:18,940 --> 00:34:23,370 So no matter what kind of data you have, 630 00:34:23,370 --> 00:34:27,774 you should be able to come up with this on your own. 631 00:34:27,774 --> 00:34:33,510 And what else is going to be happening? 632 00:34:33,510 --> 00:34:38,139 When I solve for x of-- the point corresponding to t 633 00:34:38,139 --> 00:34:39,389 equals 0. 634 00:34:39,389 --> 00:34:43,920 x of 0 and y of 0 will therefore be what? 635 00:34:43,920 --> 00:34:49,350 It will be R plus x0. 636 00:34:49,350 --> 00:34:51,429 This is going to be what? 637 00:34:51,429 --> 00:34:53,710 Just y0. 638 00:34:53,710 --> 00:34:56,194 Does anybody give them to me? 639 00:34:56,194 --> 00:34:59,110 STUDENT: 3, 0. 640 00:34:59,110 --> 00:35:01,630 PROFESSOR TODA: Alexander gave me the correct ones. 641 00:35:01,630 --> 00:35:05,670 They will be 3 and 0. 642 00:35:05,670 --> 00:35:06,670 Are you guys with me? 643 00:35:06,670 --> 00:35:10,920 They could be anything, anything that makes sense. 644 00:35:10,920 --> 00:35:15,100 All right, for example somebody would say, I'm starting here. 645 00:35:15,100 --> 00:35:16,580 I give you other points. 646 00:35:16,580 --> 00:35:19,910 Then you put them in, you plug in that initial point, 647 00:35:19,910 --> 00:35:22,980 meaning that you're starting your motion here. 648 00:35:22,980 --> 00:35:26,120 And you do go around the circle one 649 00:35:26,120 --> 00:35:31,688 because, you take [INAUDIBLE] only between 0 and 2 pi. 650 00:35:31,688 --> 00:35:32,852 Alexander. 651 00:35:32,852 --> 00:35:34,018 STUDENT: I have [INAUDIBLE]. 652 00:35:34,018 --> 00:35:34,950 PROFESSOR TODA: OK. 653 00:35:34,950 --> 00:35:35,882 STUDENT: [INAUDIBLE] 654 00:35:35,882 --> 00:35:38,310 PROFESSOR TODA: No, I thought that I misprinted something 655 00:35:38,310 --> 00:35:38,610 again. 656 00:35:38,610 --> 00:35:40,784 STUDENT: No, I was about to say something really dumb. 657 00:35:40,784 --> 00:35:41,575 PROFESSOR TODA: OK. 658 00:35:41,575 --> 00:35:43,840 659 00:35:43,840 --> 00:35:48,692 So how do we make sense of what we have here? 660 00:35:48,692 --> 00:35:52,280 Well, y0 corresponds to what I said. 661 00:35:52,280 --> 00:35:55,680 So this is a superfluous equation. 662 00:35:55,680 --> 00:35:57,620 I don't need that. 663 00:35:57,620 --> 00:36:01,200 What do I know from that? 664 00:36:01,200 --> 00:36:05,824 R will be 2. 665 00:36:05,824 --> 00:36:07,690 x1 is 1. 666 00:36:07,690 --> 00:36:10,000 I have a superfluous equation. 667 00:36:10,000 --> 00:36:13,890 I have to get identities in that case, right? 668 00:36:13,890 --> 00:36:14,700 OK, now. 669 00:36:14,700 --> 00:36:19,640 670 00:36:19,640 --> 00:36:26,640 What is going to be my-- my bunch of equations 671 00:36:26,640 --> 00:36:47,640 will be x of t equals 2 cosine t plus 1 and y of t 672 00:36:47,640 --> 00:36:49,230 equals-- I don't like this marker. 673 00:36:49,230 --> 00:36:49,920 I hate it. 674 00:36:49,920 --> 00:36:50,772 Where did I get it? 675 00:36:50,772 --> 00:36:51,730 In the math department. 676 00:36:51,730 --> 00:36:53,460 And it's a new one. 677 00:36:53,460 --> 00:36:54,820 I got it as a new one. 678 00:36:54,820 --> 00:36:56,740 It's not working. 679 00:36:56,740 --> 00:36:58,180 OK, y of t. 680 00:36:58,180 --> 00:37:01,150 681 00:37:01,150 --> 00:37:03,830 The blue contrast is invisible. 682 00:37:03,830 --> 00:37:07,897 I have 2 sine t. 683 00:37:07,897 --> 00:37:08,396 Okey dokey. 684 00:37:08,396 --> 00:37:11,590 When you finish a problem, always quickly 685 00:37:11,590 --> 00:37:15,742 verify if what you got makes sense. 686 00:37:15,742 --> 00:37:19,500 And obviously if I look at everything, 687 00:37:19,500 --> 00:37:21,130 it's matching the whole point. 688 00:37:21,130 --> 00:37:21,906 Right? 689 00:37:21,906 --> 00:37:22,680 OK. 690 00:37:22,680 --> 00:37:29,843 Now going back to-- this is reminding me of something in 3d 691 00:37:29,843 --> 00:37:34,673 that I wanted to talk to you today about. 692 00:37:34,673 --> 00:37:36,605 This is [INAUDIBLE]. 693 00:37:36,605 --> 00:37:42,900 694 00:37:42,900 --> 00:37:45,120 I'm going to give you, in a similar way 695 00:37:45,120 --> 00:37:47,501 with this simple problem, I'm going 696 00:37:47,501 --> 00:37:49,886 to give you something more complicated 697 00:37:49,886 --> 00:38:16,700 and say find the parametrization of a helix. 698 00:38:16,700 --> 00:38:19,610 And you say, well, I'm happy that this 699 00:38:19,610 --> 00:38:21,824 isn't a made-up problem again. 700 00:38:21,824 --> 00:38:23,918 I have to be a little bit more careful 701 00:38:23,918 --> 00:38:27,430 with these made-up problems so that they make sense. 702 00:38:27,430 --> 00:38:44,380 Of a helix R of t such that it is contained or it lies, 703 00:38:44,380 --> 00:38:59,770 it lies on the circular cylinder x squared 704 00:38:59,770 --> 00:39:03,822 plus y squared equals 4. 705 00:39:03,822 --> 00:39:04,780 Why is that a cylinder? 706 00:39:04,780 --> 00:39:07,910 The z's missing, so it's going to be a cylinder whose 707 00:39:07,910 --> 00:39:09,470 main axis is the z axis. 708 00:39:09,470 --> 00:39:09,970 Right? 709 00:39:09,970 --> 00:39:11,450 Are you guys with me? 710 00:39:11,450 --> 00:39:14,750 I think we are on the same page. 711 00:39:14,750 --> 00:39:19,449 And you cannot solve the problem just with this data. 712 00:39:19,449 --> 00:39:22,160 Do you agree with me? 713 00:39:22,160 --> 00:39:47,190 And knowing that, the curvature of the helix is k 714 00:39:47,190 --> 00:40:04,030 equals 2/5 at every point. 715 00:40:04,030 --> 00:40:06,080 And of course it's an oxymoron. 716 00:40:06,080 --> 00:40:08,240 Because what I proved last time is 717 00:40:08,240 --> 00:40:12,530 that the curvature of a helix is a constant. 718 00:40:12,530 --> 00:40:27,220 So remember, we got the curvature of a helix 719 00:40:27,220 --> 00:40:30,056 as being a constant. 720 00:40:30,056 --> 00:40:34,455 721 00:40:34,455 --> 00:40:36,413 STUDENT: What's that last word of the sentence? 722 00:40:36,413 --> 00:40:38,589 It's "the curvature is at every" what? 723 00:40:38,589 --> 00:40:39,880 PROFESSOR TODA: At every point. 724 00:40:39,880 --> 00:40:45,070 I'm sorry I said, it very-- I abbreviated [INAUDIBLE]. 725 00:40:45,070 --> 00:40:48,100 So at every point you have the same curvature. 726 00:40:48,100 --> 00:40:50,920 When you draw a helix you say, wait, 727 00:40:50,920 --> 00:40:53,820 the helix is bent uniformly. 728 00:40:53,820 --> 00:40:58,760 If you were to play with a spring taken from am old bed, 729 00:40:58,760 --> 00:41:01,910 you would go with your hands along the spring. 730 00:41:01,910 --> 00:41:04,700 And then you say, oh, it bends about the same. 731 00:41:04,700 --> 00:41:06,010 Yes, it does. 732 00:41:06,010 --> 00:41:08,800 And that means the curvature is the same. 733 00:41:08,800 --> 00:41:11,720 How would you solve this problem? 734 00:41:11,720 --> 00:41:16,380 This problem is hard, because you cannot integrate 735 00:41:16,380 --> 00:41:17,450 the curvature. 736 00:41:17,450 --> 00:41:19,110 Well, what is the curvature? 737 00:41:19,110 --> 00:41:21,040 The curvature would be-- 738 00:41:21,040 --> 00:41:22,040 STUDENT: Absolute value. 739 00:41:22,040 --> 00:41:23,910 PROFESSOR TODA: Just absolute value of R 740 00:41:23,910 --> 00:41:28,410 double prime if it were in s. 741 00:41:28,410 --> 00:41:31,030 And you cannot integrate. 742 00:41:31,030 --> 00:41:34,000 If somebody gave you the vector equation 743 00:41:34,000 --> 00:41:36,570 of double prime of this, them you say, 744 00:41:36,570 --> 00:41:38,730 yes, I can integrate one step going back. 745 00:41:38,730 --> 00:41:40,330 I get R prime of s. 746 00:41:40,330 --> 00:41:41,729 Then I go back to R of s. 747 00:41:41,729 --> 00:41:43,270 But this is a little bit complicated. 748 00:41:43,270 --> 00:41:45,492 I'm giving you a scalar. 749 00:41:45,492 --> 00:41:50,760 You have to be a little bit aware of what you did last time 750 00:41:50,760 --> 00:41:54,596 and try to remember what we did last time. 751 00:41:54,596 --> 00:41:56,330 What did we do last time? 752 00:41:56,330 --> 00:41:58,110 I would not give you a problem like that 753 00:41:58,110 --> 00:42:03,290 on the final, because it would assume that you have solved 754 00:42:03,290 --> 00:42:06,282 the problem we did last time in terms of R of t 755 00:42:06,282 --> 00:42:09,950 equals A equals sine t. 756 00:42:09,950 --> 00:42:11,400 A sine t and [? vt. ?] 757 00:42:11,400 --> 00:42:15,770 And we said, this is the standard parametrized helix 758 00:42:15,770 --> 00:42:22,330 that sits on a cylinder of radius A and has the phb. 759 00:42:22,330 --> 00:42:27,510 So the distance between consecutive spirals 760 00:42:27,510 --> 00:42:28,770 really matters. 761 00:42:28,770 --> 00:42:30,160 That really makes the difference. 762 00:42:30,160 --> 00:42:30,620 STUDENT: I have a question. 763 00:42:30,620 --> 00:42:32,578 PROFESSOR TODA: You wanted to ask me something. 764 00:42:32,578 --> 00:42:34,350 STUDENT: Is s always the reciprocal of t? 765 00:42:34,350 --> 00:42:35,952 Are they always-- 766 00:42:35,952 --> 00:42:37,410 PROFESSOR TODA: No, not reciprocal. 767 00:42:37,410 --> 00:42:45,800 You mean s of t is a function is from t0 to t of the speed. 768 00:42:45,800 --> 00:42:50,430 R prime and t-- d tau, right? 769 00:42:50,430 --> 00:42:51,630 Tau not t. [INAUDIBLE]. 770 00:42:51,630 --> 00:42:54,220 771 00:42:54,220 --> 00:43:00,650 t and s are different parameters. 772 00:43:00,650 --> 00:43:01,993 Different times. 773 00:43:01,993 --> 00:43:04,369 Different parameter times. 774 00:43:04,369 --> 00:43:04,910 And you say-- 775 00:43:04,910 --> 00:43:06,701 STUDENT: Isn't s the parameter time 776 00:43:06,701 --> 00:43:08,807 when [INAUDIBLE] parametrized? 777 00:43:08,807 --> 00:43:09,890 PROFESSOR TODA: Very good. 778 00:43:09,890 --> 00:43:12,365 So what is the magic s? 779 00:43:12,365 --> 00:43:13,850 I'm proud of you. 780 00:43:13,850 --> 00:43:15,940 This is the important thing to remember. 781 00:43:15,940 --> 00:43:17,530 t could be any time. 782 00:43:17,530 --> 00:43:19,960 I start measuring wherever I want. 783 00:43:19,960 --> 00:43:23,690 I can set my watch to start now. 784 00:43:23,690 --> 00:43:24,950 It could be crazy. 785 00:43:24,950 --> 00:43:26,640 Doesn't have to be uniform. 786 00:43:26,640 --> 00:43:27,603 Motion, I don't care. 787 00:43:27,603 --> 00:43:30,760 788 00:43:30,760 --> 00:43:33,330 s is a friend of yours that says, 789 00:43:33,330 --> 00:43:38,290 I am that special time so that according to me 790 00:43:38,290 --> 00:43:40,590 the speed will become one. 791 00:43:40,590 --> 00:43:45,650 So for a physicist to measure the speed with respect to this, 792 00:43:45,650 --> 00:43:49,410 parameter s time, the speed will always become one. 793 00:43:49,410 --> 00:43:51,660 That is the arclength time and position. 794 00:43:51,660 --> 00:43:54,480 How you get from one another, I told you last time 795 00:43:54,480 --> 00:43:57,376 that for both of them you have-- this is R of t 796 00:43:57,376 --> 00:43:59,230 and this is little r of s. 797 00:43:59,230 --> 00:44:01,240 And there is a composition. 798 00:44:01,240 --> 00:44:03,460 s can be viewed as a function of t, 799 00:44:03,460 --> 00:44:06,340 and t can be viewed as a function of s. 800 00:44:06,340 --> 00:44:09,580 As functions they are inverse to one another. 801 00:44:09,580 --> 00:44:12,650 So going back to who they are, a very good question, 802 00:44:12,650 --> 00:44:15,580 because this is a review anyway, [? who wants ?] 803 00:44:15,580 --> 00:44:19,180 s as a function of t for this particular problem? 804 00:44:19,180 --> 00:44:23,949 I hope you remember, we were like-- have you seen this movie 805 00:44:23,949 --> 00:44:28,395 with Mickey Mouse going on a mountain that 806 00:44:28,395 --> 00:44:32,100 was more like a cylinder. 807 00:44:32,100 --> 00:44:35,320 And this is the train going at a constant slope. 808 00:44:35,320 --> 00:44:42,590 And one of my colleagues, actually, he's at Stanford, 809 00:44:42,590 --> 00:44:47,300 was telling me that he gave his students in Calc 1 810 00:44:47,300 --> 00:44:51,860 to prove, formally prove, that the angle formed 811 00:44:51,860 --> 00:44:56,590 by the law of motion by the velocity vector, 812 00:44:56,590 --> 00:45:01,990 with the horizontal plane passing through the particle, 813 00:45:01,990 --> 00:45:04,050 is always a constant. 814 00:45:04,050 --> 00:45:07,165 I didn't think about doing in now, but of course we can. 815 00:45:07,165 --> 00:45:08,520 We could do that. 816 00:45:08,520 --> 00:45:10,964 So maybe the next thing would be, like, 817 00:45:10,964 --> 00:45:12,630 if you [INAUDIBLE] an extra problem, can 818 00:45:12,630 --> 00:45:17,280 we show that angle between the velocity vector on the helix 819 00:45:17,280 --> 00:45:20,702 and the horizontal plane through that point is a constant. 820 00:45:20,702 --> 00:45:22,535 STUDENT: Wouldn't it just be, because B of t 821 00:45:22,535 --> 00:45:23,935 is just a constant times t? 822 00:45:23,935 --> 00:45:24,810 PROFESSOR TODA: Yeah. 823 00:45:24,810 --> 00:45:25,590 We'll get to that. 824 00:45:25,590 --> 00:45:27,170 We'll get to that in a second. 825 00:45:27,170 --> 00:45:32,487 So he reminded me of an old movie from like 70 years ago, 826 00:45:32,487 --> 00:45:33,820 with Mickey Mouse and the train. 827 00:45:33,820 --> 00:45:38,650 And the train going up at the same speed. 828 00:45:38,650 --> 00:45:41,160 You have to maintain the same speed. 829 00:45:41,160 --> 00:45:44,810 Because if you risk it not, then you sort of 830 00:45:44,810 --> 00:45:46,160 are getting trouble. 831 00:45:46,160 --> 00:45:47,760 So you never stop. 832 00:45:47,760 --> 00:45:49,290 If you stop you go back. 833 00:45:49,290 --> 00:45:50,740 So it's a regular curve. 834 00:45:50,740 --> 00:45:52,875 What I have here is that such a curve. 835 00:45:52,875 --> 00:45:54,786 Regular curve, never stop. 836 00:45:54,786 --> 00:45:56,800 Get up with a constant speed. 837 00:45:56,800 --> 00:45:58,826 Do you guys remember the speed from last time? 838 00:45:58,826 --> 00:46:01,077 We'll square root the a squared plus b squared. 839 00:46:01,077 --> 00:46:04,430 When we did the velocity thingie. 840 00:46:04,430 --> 00:46:10,730 And I get square root a squared plus b squared times t. 841 00:46:10,730 --> 00:46:19,040 Now, today I would like to ask you one question. 842 00:46:19,040 --> 00:46:21,520 What if-- Ryan brought this up. 843 00:46:21,520 --> 00:46:22,460 It's very good. 844 00:46:22,460 --> 00:46:23,660 b is a constant. 845 00:46:23,660 --> 00:46:26,550 What if b would not be a constant, 846 00:46:26,550 --> 00:46:28,610 or maybe could be worse? 847 00:46:28,610 --> 00:46:32,710 For example, instead of having another linear function with t, 848 00:46:32,710 --> 00:46:36,178 but something that contains higher powers of t. 849 00:46:36,178 --> 00:46:39,360 850 00:46:39,360 --> 00:46:43,410 Then you don't go at the constant speed anymore. 851 00:46:43,410 --> 00:46:45,370 You can say goodbye to the cartoon. 852 00:46:45,370 --> 00:46:45,880 Yes, sir? 853 00:46:45,880 --> 00:46:49,017 STUDENT: And then it's [INAUDIBLE]. 854 00:46:49,017 --> 00:46:50,100 One that goes [INAUDIBLE]. 855 00:46:50,100 --> 00:46:51,600 PROFESSOR TODA: I mean, it's still-- 856 00:46:51,600 --> 00:46:54,826 STUDENT: s is not multiplied by a constant. 857 00:46:54,826 --> 00:46:57,017 The function between t and s is not a constant one. 858 00:46:57,017 --> 00:46:59,600 PROFESSOR TODA: It's going to be a different parameterization, 859 00:46:59,600 --> 00:47:00,580 different speed. 860 00:47:00,580 --> 00:47:03,770 Sometimes-- OK, you have to understand. 861 00:47:03,770 --> 00:47:06,740 Let's say I have a cone. 862 00:47:06,740 --> 00:47:10,230 And I'm going slow at first, and I 863 00:47:10,230 --> 00:47:11,980 go faster and faster and faster and faster 864 00:47:11,980 --> 00:47:13,900 to the end of the cone. 865 00:47:13,900 --> 00:47:18,390 But then I have the same physical curve, 866 00:47:18,390 --> 00:47:21,040 and I parameterized [INAUDIBLE] the length. 867 00:47:21,040 --> 00:47:24,310 And I say, no, I'm a mechanic. 868 00:47:24,310 --> 00:47:26,840 Or I'm the engineer of the strain. 869 00:47:26,840 --> 00:47:29,420 I can make the motion have a constant speed. 870 00:47:29,420 --> 00:47:33,130 So even if the helix is no longer circular, 871 00:47:33,130 --> 00:47:36,560 and it's sort of a crazy helix going on top of the mountain, 872 00:47:36,560 --> 00:47:39,330 as an engineer I can just say, oh no, 873 00:47:39,330 --> 00:47:42,150 I want cruise control for my little train. 874 00:47:42,150 --> 00:47:45,500 And I will go at the same speed. 875 00:47:45,500 --> 00:47:48,940 See, the problem is the slope a constant. 876 00:47:48,940 --> 00:47:51,270 And thinking of what they did that 877 00:47:51,270 --> 00:47:53,067 stand for, because it didn't stand 878 00:47:53,067 --> 00:47:54,880 for [INAUDIBLE] in honors. 879 00:47:54,880 --> 00:47:57,280 We can do it in honors as well. 880 00:47:57,280 --> 00:47:58,700 We'll do it in a second. 881 00:47:58,700 --> 00:48:04,950 Now, k obviously is what? 882 00:48:04,950 --> 00:48:08,460 Some of you have very good memory, 883 00:48:08,460 --> 00:48:13,250 and like the memory of a medical doctor, which is great. 884 00:48:13,250 --> 00:48:14,560 Some of you don't. 885 00:48:14,560 --> 00:48:18,772 But if you don't you just go back and look at the notes. 886 00:48:18,772 --> 00:48:20,756 What I'm trying to do, but I don't know, 887 00:48:20,756 --> 00:48:22,960 it's also a matter of money-- I don't 888 00:48:22,960 --> 00:48:26,040 want to use the math department copier-- I'd 889 00:48:26,040 --> 00:48:29,850 like to make a stack of notes. 890 00:48:29,850 --> 00:48:33,090 So that's why I'm collecting these notes, to bring them back 891 00:48:33,090 --> 00:48:33,971 to you. 892 00:48:33,971 --> 00:48:34,470 For free! 893 00:48:34,470 --> 00:48:36,470 I'm not going to sell them to you. 894 00:48:36,470 --> 00:48:38,060 I'm [INAUDIBLE]. 895 00:48:38,060 --> 00:48:41,515 So that you can have those with you whenever you want, 896 00:48:41,515 --> 00:48:45,475 or put them in a spiral, punch holes in them, 897 00:48:45,475 --> 00:48:48,680 and have them for review at any time. 898 00:48:48,680 --> 00:48:51,450 Reminds me of what that was-- that was in the notes. 899 00:48:51,450 --> 00:48:54,694 a over a squared plus b squared. 900 00:48:54,694 --> 00:48:57,300 So who can tell me, a and b really quickly, 901 00:48:57,300 --> 00:49:00,870 so we don't waste too much time, Mr. a is--? 902 00:49:00,870 --> 00:49:05,729 903 00:49:05,729 --> 00:49:07,020 STUDENT: So this is another way 904 00:49:07,020 --> 00:49:07,601 STUDENT: 2. 905 00:49:07,601 --> 00:49:08,350 PROFESSOR TODA: 2. 906 00:49:08,350 --> 00:49:13,037 STUDENT: So is this another way of defining k in k of s? 907 00:49:13,037 --> 00:49:14,120 PROFESSOR TODA: Actually-- 908 00:49:14,120 --> 00:49:16,895 STUDENT: That's the general curvature for [INAUDIBLE]. 909 00:49:16,895 --> 00:49:21,040 PROFESSOR TODA: You know what is the magic thing? 910 00:49:21,040 --> 00:49:23,095 Even if-- the curvature is an invariant. 911 00:49:23,095 --> 00:49:26,526 It doesn't depend the reparametrization. 912 00:49:26,526 --> 00:49:29,830 There is a way maybe I'm going to teach you, although this 913 00:49:29,830 --> 00:49:32,160 is not in the book. 914 00:49:32,160 --> 00:49:35,870 What are the formulas corresponding 915 00:49:35,870 --> 00:49:41,580 to the [INAUDIBLE] t and v that depend on curvature and torsion 916 00:49:41,580 --> 00:49:44,000 and the speed along the curve. 917 00:49:44,000 --> 00:49:48,750 And if you analyze the notion of curvature, [INAUDIBLE], 918 00:49:48,750 --> 00:49:52,230 no matter what your parameter will be, t, s, tau, 919 00:49:52,230 --> 00:49:56,690 God knows what, k will still be the same number. 920 00:49:56,690 --> 00:49:59,300 So k is viewed as an invariant with respect 921 00:49:59,300 --> 00:50:01,425 to the parametrization. 922 00:50:01,425 --> 00:50:04,120 STUDENT: So then that a over a squared plus b squared, 923 00:50:04,120 --> 00:50:05,912 that's another way of finding k? 924 00:50:05,912 --> 00:50:07,120 PROFESSOR TODA: Say it again? 925 00:50:07,120 --> 00:50:09,285 STUDENT: So using a over a squared plus b squared 926 00:50:09,285 --> 00:50:10,829 is another way of finding k? 927 00:50:10,829 --> 00:50:11,620 PROFESSOR TODA: No. 928 00:50:11,620 --> 00:50:13,916 Somebody gave you k. 929 00:50:13,916 --> 00:50:17,440 And then you say, if it's a standard parametrization, 930 00:50:17,440 --> 00:50:25,290 and then I get 2/5, can I be sure a is 2? 931 00:50:25,290 --> 00:50:28,220 I'm sure a is 2 from nothing. 932 00:50:28,220 --> 00:50:32,860 This is what makes me aware that a is 2 the first place. 933 00:50:32,860 --> 00:50:36,640 Because its the radius of the cylinder. 934 00:50:36,640 --> 00:50:39,290 This is x squared, x and y. 935 00:50:39,290 --> 00:50:41,860 You see, x squared plus y squared is a squared. 936 00:50:41,860 --> 00:50:43,650 This is where I get a from. 937 00:50:43,650 --> 00:50:44,620 a is 2. 938 00:50:44,620 --> 00:50:47,140 I replace it in here and I say, all righty, 939 00:50:47,140 --> 00:50:51,777 so I only have one choice. a is 2 and b is? 940 00:50:51,777 --> 00:50:52,610 STUDENT: [INAUDIBLE] 941 00:50:52,610 --> 00:50:57,260 942 00:50:57,260 --> 00:51:00,470 PROFESSOR TODA: But can b plus-- So what I'm saying, 943 00:51:00,470 --> 00:51:01,380 a is 2, right? 944 00:51:01,380 --> 00:51:04,390 We know that from this. 945 00:51:04,390 --> 00:51:08,440 If I block in here I have 4 and somebody says plus minus 1. 946 00:51:08,440 --> 00:51:09,520 No. 947 00:51:09,520 --> 00:51:11,000 b is always positive. 948 00:51:11,000 --> 00:51:13,380 So you remember the last time we discussed 949 00:51:13,380 --> 00:51:16,640 about the standard parametrization. 950 00:51:16,640 --> 00:51:20,280 But somebody will say, but what if I put a minus? 951 00:51:20,280 --> 00:51:22,840 What if I'm going to put a minus? 952 00:51:22,840 --> 00:51:24,150 That's an excellent question. 953 00:51:24,150 --> 00:51:27,072 What's going to happen if you put minus t? 954 00:51:27,072 --> 00:51:28,010 [INTERPOSING VOICES] 955 00:51:28,010 --> 00:51:29,010 PROFESSOR TODA: Exactly. 956 00:51:29,010 --> 00:51:31,260 In the opposite direction. 957 00:51:31,260 --> 00:51:35,550 Instead of going up, you go down. 958 00:51:35,550 --> 00:51:37,430 All right. 959 00:51:37,430 --> 00:51:41,095 Now, I'm gonna-- what else? 960 00:51:41,095 --> 00:51:43,270 Ah, I said, let's do this. 961 00:51:43,270 --> 00:51:47,986 Let's prove that the angle is a constant, 962 00:51:47,986 --> 00:51:51,080 the angle that's made by the velocity 963 00:51:51,080 --> 00:51:56,220 vector of the train with the horizontal plane is a constant. 964 00:51:56,220 --> 00:51:57,840 Is this hard? 965 00:51:57,840 --> 00:51:58,340 Nah. 966 00:51:58,340 --> 00:51:58,840 Yes, sir? 967 00:51:58,840 --> 00:52:03,930 STUDENT: Are we still going to find R of t given only k? 968 00:52:03,930 --> 00:52:05,550 PROFESSOR TODA: But didn't we? 969 00:52:05,550 --> 00:52:07,300 We did. 970 00:52:07,300 --> 00:52:13,750 R of t was 2 cosine t, 2 sine t, and t. 971 00:52:13,750 --> 00:52:16,280 All right? 972 00:52:16,280 --> 00:52:17,470 OK, so we are done. 973 00:52:17,470 --> 00:52:18,640 What did I say? 974 00:52:18,640 --> 00:52:22,410 I said that let's prove-- it's a proof. 975 00:52:22,410 --> 00:52:27,305 Let's prove that the angle made by the velocity to the train-- 976 00:52:27,305 --> 00:52:30,635 to the train?-- to the direction of motion, which is the helix. 977 00:52:30,635 --> 00:52:37,438 And the horizontal plane is a constant. 978 00:52:37,438 --> 00:52:38,426 Is this hard? 979 00:52:38,426 --> 00:52:39,908 How are we going to do that? 980 00:52:39,908 --> 00:52:42,872 Now I start waking up, because I was very tired. 981 00:52:42,872 --> 00:52:44,259 STUDENT: [INAUDIBLE] 982 00:52:44,259 --> 00:52:45,342 PROFESSOR TODA: Excuse me. 983 00:52:45,342 --> 00:52:46,840 STUDENT: [INAUDIBLE] 984 00:52:46,840 --> 00:53:01,242 PROFESSOR TODA: So you see, the helix contains this point. 985 00:53:01,242 --> 00:53:03,920 And I'm looking at the velocity vector 986 00:53:03,920 --> 00:53:06,310 that is standard to the helix. 987 00:53:06,310 --> 00:53:09,320 And I'll call that R prime. 988 00:53:09,320 --> 00:53:10,980 And then you say, yea, but how am I 989 00:53:10,980 --> 00:53:13,990 going to compute that angle? 990 00:53:13,990 --> 00:53:15,632 What is that angle? 991 00:53:15,632 --> 00:53:17,987 STUDENT: It's a function of b. 992 00:53:17,987 --> 00:53:20,820 993 00:53:20,820 --> 00:53:21,980 PROFESSOR TODA: It will be. 994 00:53:21,980 --> 00:53:24,820 But we have to do it rigorously. 995 00:53:24,820 --> 00:53:27,925 So what's going to happen for me to draw that angle? 996 00:53:27,925 --> 00:53:30,094 First of all, I should take-- from the tip 997 00:53:30,094 --> 00:53:33,240 of the vector I should draw perpendicular 998 00:53:33,240 --> 00:53:36,235 to the horizontal plane passing through the point. 999 00:53:36,235 --> 00:53:37,110 And I'll get P prime. 1000 00:53:37,110 --> 00:53:37,693 God knows why. 1001 00:53:37,693 --> 00:53:41,490 I don't know why, I don't know why. [? Q. ?] And this is PR, 1002 00:53:41,490 --> 00:53:42,910 and P-- not PR. 1003 00:53:42,910 --> 00:53:46,990 PR is too much [INAUDIBLE] radius, M. 1004 00:53:46,990 --> 00:53:50,835 OK, so then you would take PQ and then 1005 00:53:50,835 --> 00:53:52,604 you would measure this angle. 1006 00:53:52,604 --> 00:53:54,770 Well, you have to be a little bit smarter than that, 1007 00:53:54,770 --> 00:53:58,390 because you can prove something else. 1008 00:53:58,390 --> 00:54:02,930 This is the complement of another angle that you love. 1009 00:54:02,930 --> 00:54:07,095 And using chapter 9 you can do that angle in no time. 1010 00:54:07,095 --> 00:54:15,840 1011 00:54:15,840 --> 00:54:20,800 So this is the complement of the angle 1012 00:54:20,800 --> 00:54:23,500 formed by the velocity vector of prime with the normal. 1013 00:54:23,500 --> 00:54:26,680 1014 00:54:26,680 --> 00:54:29,720 But not the normal principle normal to the curve, 1015 00:54:29,720 --> 00:54:32,340 but the normal to the plane. 1016 00:54:32,340 --> 00:54:34,510 And what is the normal to the plane? 1017 00:54:34,510 --> 00:54:38,960 Let's call the principal normal n to the curve big N bar. 1018 00:54:38,960 --> 00:54:42,110 So in order to avoid confusion, I'll write this little n. 1019 00:54:42,110 --> 00:54:42,945 How about that? 1020 00:54:42,945 --> 00:54:45,240 Do you guys know-- like they do in mechanics. 1021 00:54:45,240 --> 00:54:48,360 If you have two normals, they call that 1n. 1022 00:54:48,360 --> 00:54:51,200 1 is little n, and stuff like that. 1023 00:54:51,200 --> 00:54:52,980 So this is the complement. 1024 00:54:52,980 --> 00:54:55,430 If I were able to prove that that complement 1025 00:54:55,430 --> 00:54:59,600 is a constant-- this is the Stanford [? property-- ?] then 1026 00:54:59,600 --> 00:55:00,990 I will be happy. 1027 00:55:00,990 --> 00:55:03,100 Is it hard? 1028 00:55:03,100 --> 00:55:04,286 No, for god's sake. 1029 00:55:04,286 --> 00:55:07,034 Who is little n? 1030 00:55:07,034 --> 00:55:11,350 Little n would be-- is that the normal to a plane 1031 00:55:11,350 --> 00:55:12,330 that you love? 1032 00:55:12,330 --> 00:55:13,340 What is your plane? 1033 00:55:13,340 --> 00:55:14,090 STUDENT: xy plane. 1034 00:55:14,090 --> 00:55:16,520 PROFESSOR TODA: Your plane is horizontal plane. 1035 00:55:16,520 --> 00:55:17,320 STUDENT: xy. 1036 00:55:17,320 --> 00:55:18,570 PROFESSOR TODA: Yes, xy plane. 1037 00:55:18,570 --> 00:55:22,120 Or xy plane shifted, shifted, shifted, shifted. 1038 00:55:22,120 --> 00:55:23,180 That's the normal change? 1039 00:55:23,180 --> 00:55:23,679 No. 1040 00:55:23,679 --> 00:55:24,886 Who is the normal? 1041 00:55:24,886 --> 00:55:26,174 STUDENT: [INAUDIBLE] 1042 00:55:26,174 --> 00:55:27,340 PROFESSOR TODA: [INAUDIBLE]. 1043 00:55:27,340 --> 00:55:28,308 STUDENT: 0, 0, 1. 1044 00:55:28,308 --> 00:55:29,308 PROFESSOR TODA: 0, 0, 1. 1045 00:55:29,308 --> 00:55:29,808 OK. 1046 00:55:29,808 --> 00:55:32,110 When I put 0 I was [INAUDIBLE]. 1047 00:55:32,110 --> 00:55:33,720 So this is k. 1048 00:55:33,720 --> 00:55:36,420 1049 00:55:36,420 --> 00:55:37,730 All right. 1050 00:55:37,730 --> 00:55:39,620 And what is our prime? 1051 00:55:39,620 --> 00:55:42,300 I was-- that was a piece of cake. 1052 00:55:42,300 --> 00:55:47,360 We did it last time minus a sine t, a equals sine t and b. 1053 00:55:47,360 --> 00:55:50,720 1054 00:55:50,720 --> 00:55:53,620 Let's find that angle. 1055 00:55:53,620 --> 00:55:54,690 Well, I don't know. 1056 00:55:54,690 --> 00:55:58,320 You have to teach me, because you have chapter 9 fresher 1057 00:55:58,320 --> 00:56:01,590 in your memory than I have it. 1058 00:56:01,590 --> 00:56:03,920 Are you taking attendance also? 1059 00:56:03,920 --> 00:56:07,177 Are you writing your name down? 1060 00:56:07,177 --> 00:56:08,260 Oh, no problem whatsoever. 1061 00:56:08,260 --> 00:56:09,391 STUDENT: We didn't get it. 1062 00:56:09,391 --> 00:56:10,807 PROFESSOR TODA: You didn't get it. 1063 00:56:10,807 --> 00:56:11,750 Circulate it. 1064 00:56:11,750 --> 00:56:16,660 All right, so who is going to help me with the angle? 1065 00:56:16,660 --> 00:56:19,870 What is the angle between two vectors, guys? 1066 00:56:19,870 --> 00:56:24,070 That should be review from what we just covered in chapter 9. 1067 00:56:24,070 --> 00:56:27,980 Let me call them u and v. And who's 1068 00:56:27,980 --> 00:56:29,916 going to tell me how I get that angle? 1069 00:56:29,916 --> 00:56:31,960 STUDENT: [INAUDIBLE] is equal to the inverse cosine of the dot 1070 00:56:31,960 --> 00:56:33,290 product of [? the magnitude. ?] 1071 00:56:33,290 --> 00:56:35,081 PROFESSOR TODA: Do you like me to write arc 1072 00:56:35,081 --> 00:56:36,450 cosine or cosine [INAUDIBLE]. 1073 00:56:36,450 --> 00:56:37,760 Doesn't matter. 1074 00:56:37,760 --> 00:56:39,850 Arc cosine of-- 1075 00:56:39,850 --> 00:56:40,960 STUDENT: The dot products. 1076 00:56:40,960 --> 00:56:47,476 PROFESSOR TODA: The dot product between u and v. 1077 00:56:47,476 --> 00:56:48,940 STUDENT: Over magnitude. 1078 00:56:48,940 --> 00:56:52,466 PROFESSOR TODA: Divided by the product of their magnitudes. 1079 00:56:52,466 --> 00:56:54,692 Look, I will change the order, because you're not 1080 00:56:54,692 --> 00:56:56,140 going to like it. 1081 00:56:56,140 --> 00:56:56,910 Doesn't matter. 1082 00:56:56,910 --> 00:56:57,680 OK? 1083 00:56:57,680 --> 00:57:03,450 So the angle phi between my favorite vectors 1084 00:57:03,450 --> 00:57:08,460 here is going to be simply the dot product. 1085 00:57:08,460 --> 00:57:09,570 That's a blessing. 1086 00:57:09,570 --> 00:57:10,242 It's a constant. 1087 00:57:10,242 --> 00:57:11,908 STUDENT: So you're doing the dot product 1088 00:57:11,908 --> 00:57:13,416 between the normal [INAUDIBLE]? 1089 00:57:13,416 --> 00:57:14,999 PROFESSOR TODA: Between this and that. 1090 00:57:14,999 --> 00:57:18,065 So this is u and this is v. So the dot product 1091 00:57:18,065 --> 00:57:22,340 would be 0 plus v. So the dot product 1092 00:57:22,340 --> 00:57:28,520 is arc cosine of v, which, thank god, is a constant. 1093 00:57:28,520 --> 00:57:30,310 I don't have to do anything anymore. 1094 00:57:30,310 --> 00:57:33,154 I'm done with the proof bit, because arc cosine 1095 00:57:33,154 --> 00:57:36,000 of a constant will be a constant. 1096 00:57:36,000 --> 00:57:36,720 OK? 1097 00:57:36,720 --> 00:57:37,600 All right. 1098 00:57:37,600 --> 00:57:40,850 So I have v over what? 1099 00:57:40,850 --> 00:57:45,090 What is the length of this vector? 1100 00:57:45,090 --> 00:57:46,750 1. [INAUDIBLE]. 1101 00:57:46,750 --> 00:57:50,510 What's the length of that vector? 1102 00:57:50,510 --> 00:57:55,900 Square root of a squared plus b squared. 1103 00:57:55,900 --> 00:57:56,430 All right? 1104 00:57:56,430 --> 00:58:01,831 1105 00:58:01,831 --> 00:58:05,280 STUDENT: How did you [INAUDIBLE]. 1106 00:58:05,280 --> 00:58:07,647 PROFESSOR TODA: So now let me ask you one thing. 1107 00:58:07,647 --> 00:58:11,362 1108 00:58:11,362 --> 00:58:13,910 What kind of function is arc cosine? 1109 00:58:13,910 --> 00:58:16,430 Of course I said arc cosine of a constant is a constant. 1110 00:58:16,430 --> 00:58:18,390 What kind of a function is arc cosine? 1111 00:58:18,390 --> 00:58:21,740 I'm doing review with you because I think it's useful. 1112 00:58:21,740 --> 00:58:26,068 Arc cosine is defined on what with values in what? 1113 00:58:26,068 --> 00:58:30,289 1114 00:58:30,289 --> 00:58:32,640 STUDENT: Repeat the question? 1115 00:58:32,640 --> 00:58:33,890 PROFESSOR TODA: Arc cosine. 1116 00:58:33,890 --> 00:58:36,100 Or cosine inverse, like Ryan prefers. 1117 00:58:36,100 --> 00:58:38,130 Cosine inverse is the same thing. 1118 00:58:38,130 --> 00:58:40,440 It's a function defined by where to where? 1119 00:58:40,440 --> 00:58:43,190 Cosine is defined from where to where? 1120 00:58:43,190 --> 00:58:46,136 From R to minus 1. 1121 00:58:46,136 --> 00:58:47,740 It's a cosine of t. 1122 00:58:47,740 --> 00:58:49,690 t could be any real number. 1123 00:58:49,690 --> 00:58:51,800 The range is minus 1, 1. 1124 00:58:51,800 --> 00:58:53,242 Close the interval. 1125 00:58:53,242 --> 00:58:54,950 STUDENT: So it's-- so I just wonder why-- 1126 00:58:54,950 --> 00:58:57,014 PROFESSOR TODA: Minus 1 to 1, close interval. 1127 00:58:57,014 --> 00:58:58,315 But pay attention, please. 1128 00:58:58,315 --> 00:59:03,040 Because it cannot go back to R. It has to be a 1 to 1 function. 1129 00:59:03,040 --> 00:59:05,960 You cannot have an inverse function if you don't take 1130 00:59:05,960 --> 00:59:09,242 a restriction of a function to be 1 to 1. 1131 00:59:09,242 --> 00:59:11,597 And we took that restriction of a function. 1132 00:59:11,597 --> 00:59:14,894 And do you remember what it was? 1133 00:59:14,894 --> 00:59:15,840 [INTERPOSING VOICES] 1134 00:59:15,840 --> 00:59:17,640 PROFESSOR TODA: 0 to pi. 1135 00:59:17,640 --> 00:59:19,710 Now, on this one I'm really happy. 1136 00:59:19,710 --> 00:59:23,160 Because I asked several people-- people 1137 00:59:23,160 --> 00:59:27,133 come to my office to get all sorts of transcripts, 1138 00:59:27,133 --> 00:59:27,633 [INAUDIBLE]. 1139 00:59:27,633 --> 00:59:30,600 And in trigonometry I asked one student, 1140 00:59:30,600 --> 00:59:31,910 so you took trigonometry. 1141 00:59:31,910 --> 00:59:32,910 So do you remember that? 1142 00:59:32,910 --> 00:59:34,360 He didn't remember that. 1143 00:59:34,360 --> 00:59:35,300 So I'm glad you do. 1144 00:59:35,300 --> 00:59:40,130 How about when I had the sine inverse? 1145 00:59:40,130 --> 00:59:44,760 How was my restriction so that would be a 1 to 1 function? 1146 00:59:44,760 --> 00:59:46,900 It's got to go from minus 1 to 1. 1147 00:59:46,900 --> 00:59:48,180 What is the range? 1148 00:59:48,180 --> 00:59:49,016 [INTERPOSING VOICES] 1149 00:59:49,016 --> 00:59:51,300 PROFESSOR TODA: Minus pi over 2. 1150 00:59:51,300 --> 00:59:53,253 You guys know your trig. 1151 00:59:53,253 --> 00:59:53,752 Good. 1152 00:59:53,752 --> 00:59:55,630 That's a very good thing. 1153 00:59:55,630 --> 00:59:59,440 You were in high school when you learned that? 1154 00:59:59,440 --> 01:00:00,430 Here at Lubbock High? 1155 01:00:00,430 --> 01:00:01,150 STUDENT: Yes. 1156 01:00:01,150 --> 01:00:02,066 PROFESSOR TODA: Great. 1157 01:00:02,066 --> 01:00:03,560 Good job, Lubbock High. 1158 01:00:03,560 --> 01:00:06,230 But many students, I caught them, who wanted credit 1159 01:00:06,230 --> 01:00:08,350 for trig who didn't know that. 1160 01:00:08,350 --> 01:00:09,680 Good. 1161 01:00:09,680 --> 01:00:19,870 So since arc cosine is a function that is of 0, pi, 1162 01:00:19,870 --> 01:00:25,230 for example, what if my-- let me give you an example. 1163 01:00:25,230 --> 01:00:26,910 What was last time, guys? 1164 01:00:26,910 --> 01:00:30,800 a was 1. b was 1. 1165 01:00:30,800 --> 01:00:32,010 For one example. 1166 01:00:32,010 --> 01:00:33,810 In that case, 1 with 5b. 1167 01:00:33,810 --> 01:00:36,327 STUDENT: [INAUDIBLE] ask you for the example you just did? 1168 01:00:36,327 --> 01:00:37,535 PROFESSOR TODA: No last time. 1169 01:00:37,535 --> 01:00:39,560 STUDENT: A was 3 and b was-- 1170 01:00:39,560 --> 01:00:44,255 PROFESSOR TODA: So what would that be, in this case 5? 1171 01:00:44,255 --> 01:00:46,680 STUDENT: That would be b over the square root-- 1172 01:00:46,680 --> 01:00:47,560 STUDENT: 3 over pi. 1173 01:00:47,560 --> 01:00:49,985 1174 01:00:49,985 --> 01:00:52,360 PROFESSOR TODA: a is 1 and b is 1, like we did last time. 1175 01:00:52,360 --> 01:00:55,050 STUDENT: [INAUDIBLE] 2, which is-- 1176 01:00:55,050 --> 01:00:57,059 PROFESSOR TODA: Plug in 1 is a, b is 1. 1177 01:00:57,059 --> 01:00:57,600 What is this? 1178 01:00:57,600 --> 01:00:59,417 STUDENT: It's just pi over 4. 1179 01:00:59,417 --> 01:01:00,500 PROFESSOR TODA: Pi over 4. 1180 01:01:00,500 --> 01:01:06,800 So pi will be our cosine, of 1 over square root 2, which 1181 01:01:06,800 --> 01:01:12,090 is 45 degree angle, which is-- you said pi over 4, right? 1182 01:01:12,090 --> 01:01:14,540 [INAUDIBLE]. 1183 01:01:14,540 --> 01:01:19,800 So exactly, you would have that over here. 1184 01:01:19,800 --> 01:01:22,580 This is where the cosine [INAUDIBLE]. 1185 01:01:22,580 --> 01:01:28,220 Now you see, guys, the way we have, the way I assume a and b, 1186 01:01:28,220 --> 01:01:30,980 the way anybody-- the book also introduces 1187 01:01:30,980 --> 01:01:33,350 a and b to be positive numbers. 1188 01:01:33,350 --> 01:01:37,230 Can you tell me what kind of angle phi will be, 1189 01:01:37,230 --> 01:01:39,900 not only restricted to 0 pi? 1190 01:01:39,900 --> 01:01:41,360 Well, a is positive. 1191 01:01:41,360 --> 01:01:42,480 b is positive. 1192 01:01:42,480 --> 01:01:44,360 a doesn't matter. 1193 01:01:44,360 --> 01:01:46,670 The whole thing will be positive. 1194 01:01:46,670 --> 01:01:50,510 Arc cosine of a positive number-- 1195 01:01:50,510 --> 01:01:52,010 STUDENT: Between 0 and pi over 2. 1196 01:01:52,010 --> 01:01:53,010 PROFESSOR TODA: That is. 1197 01:01:53,010 --> 01:01:56,326 Yeah, so it has to be between 0 and pi over 2. 1198 01:01:56,326 --> 01:01:57,950 So it's going to be only this quadrant. 1199 01:01:57,950 --> 01:01:59,640 Does that make sense? 1200 01:01:59,640 --> 01:02:03,388 Yes, think with the imagination of your eyes, 1201 01:02:03,388 --> 01:02:05,220 or the eyes of your imagination. 1202 01:02:05,220 --> 01:02:06,430 OK. 1203 01:02:06,430 --> 01:02:08,360 You have a cylinder. 1204 01:02:08,360 --> 01:02:10,270 And you are moving along that cylinder. 1205 01:02:10,270 --> 01:02:12,160 And this is how you turn. 1206 01:02:12,160 --> 01:02:14,400 You turn with that little train. 1207 01:02:14,400 --> 01:02:16,580 Du-du-du-du-du, you go up. 1208 01:02:16,580 --> 01:02:19,910 When you turn the velocity vector and you 1209 01:02:19,910 --> 01:02:23,123 look at the-- mm. 1210 01:02:23,123 --> 01:02:23,956 STUDENT: The normal. 1211 01:02:23,956 --> 01:02:24,860 PROFESSOR TODA: The normal! 1212 01:02:24,860 --> 01:02:25,360 Thank you. 1213 01:02:25,360 --> 01:02:30,245 The z axis, you always have an angle between 0 and pi over 2. 1214 01:02:30,245 --> 01:02:31,712 So it makes sense. 1215 01:02:31,712 --> 01:02:34,157 I'm going to go ahead and erase the whole thing. 1216 01:02:34,157 --> 01:02:41,020 1217 01:02:41,020 --> 01:02:47,720 So we reviewed, more or less, s of t, integration, derivation, 1218 01:02:47,720 --> 01:02:52,020 moving from position vector to velocity to acceleration 1219 01:02:52,020 --> 01:02:56,260 and back, acceleration to velocity to position vector, 1220 01:02:56,260 --> 01:02:58,500 the meaning of arclength. 1221 01:02:58,500 --> 01:03:00,890 There are some things I would like to tell you, 1222 01:03:00,890 --> 01:03:07,829 because Ryan asked me a few more questions about the curvature. 1223 01:03:07,829 --> 01:03:11,560 The curvature formula depends very 1224 01:03:11,560 --> 01:03:16,830 much on the type of formula you used for the curve. 1225 01:03:16,830 --> 01:03:18,800 So you say, wait, wait, wait, Magdelena, 1226 01:03:18,800 --> 01:03:21,290 you told us-- you are confusing us. 1227 01:03:21,290 --> 01:03:23,710 You told us that the curvature is uniquely 1228 01:03:23,710 --> 01:03:33,740 defined as the magnitude of the acceleration vector 1229 01:03:33,740 --> 01:03:36,800 when the law of motion is an arclength. 1230 01:03:36,800 --> 01:03:38,850 And that is correct. 1231 01:03:38,850 --> 01:03:43,190 So suppose my original law of motion was R of t [INAUDIBLE] 1232 01:03:43,190 --> 01:03:47,750 time, any time, t, any time parameter. 1233 01:03:47,750 --> 01:03:49,370 I'm making a face. 1234 01:03:49,370 --> 01:03:53,290 But then from that we switch to something beautiful, 1235 01:03:53,290 --> 01:03:56,445 which is called the arclength parametrization. 1236 01:03:56,445 --> 01:03:58,280 Why am I so happy? 1237 01:03:58,280 --> 01:04:04,970 Because in this parametrization the magnitude of the speed 1238 01:04:04,970 --> 01:04:07,120 is 1. 1239 01:04:07,120 --> 01:04:17,700 And I define k to be the magnitude 1240 01:04:17,700 --> 01:04:19,870 of R double prime of s, right? 1241 01:04:19,870 --> 01:04:22,430 The acceleration only in the arclength [? time ?] 1242 01:04:22,430 --> 01:04:23,426 parameterization. 1243 01:04:23,426 --> 01:04:24,920 And then this was the definition. 1244 01:04:24,920 --> 01:04:30,410 1245 01:04:30,410 --> 01:04:36,550 A. Can you prove-- what? 1246 01:04:36,550 --> 01:04:40,190 Can you prove the following formula? 1247 01:04:40,190 --> 01:04:52,200 1248 01:04:52,200 --> 01:04:58,514 T prime of s equals k times N of s. 1249 01:04:58,514 --> 01:05:02,550 This is famous for people who do-- not for everybody. 1250 01:05:02,550 --> 01:05:05,530 But imagine you have an engineer who does 1251 01:05:05,530 --> 01:05:08,430 research of the laws of motion. 1252 01:05:08,430 --> 01:05:13,130 Maybe he works for the railways and he's 1253 01:05:13,130 --> 01:05:17,170 looking at skew curves, or he is one 1254 01:05:17,170 --> 01:05:20,480 of those people who project the ski slopes, 1255 01:05:20,480 --> 01:05:25,360 or all sorts of winter sports slope or something, that 1256 01:05:25,360 --> 01:05:29,150 involve a lot of curvatures and torsions. 1257 01:05:29,150 --> 01:05:31,240 That guy has to know the Frenet formula. 1258 01:05:31,240 --> 01:05:34,260 So this is the famous first Frenet formula. 1259 01:05:34,260 --> 01:05:40,140 1260 01:05:40,140 --> 01:05:46,690 Frenet was a mathematician who gave the name to the TNB 1261 01:05:46,690 --> 01:05:48,476 vectors, the trihedron. 1262 01:05:48,476 --> 01:05:49,970 You have the T was what? 1263 01:05:49,970 --> 01:05:52,958 The T was the tangent [INAUDIBLE] vector. 1264 01:05:52,958 --> 01:05:58,180 The N was the principal unit normal. 1265 01:05:58,180 --> 01:06:00,930 In those videos that I'm watching that I also sent you-- 1266 01:06:00,930 --> 01:06:02,400 I like most of them. 1267 01:06:02,400 --> 01:06:05,660 I like the Khan Academy more than everything. 1268 01:06:05,660 --> 01:06:09,100 Also I like the one that was made by Dr. [? Gock ?] 1269 01:06:09,100 --> 01:06:12,540 But Dr. [? Gock ?] made a little bit of a mistake. 1270 01:06:12,540 --> 01:06:13,920 A conceptual mistake. 1271 01:06:13,920 --> 01:06:17,013 We all make mistakes by misprinting or misreading 1272 01:06:17,013 --> 01:06:18,366 or goofy mistake. 1273 01:06:18,366 --> 01:06:20,690 But he said this is the normal vector. 1274 01:06:20,690 --> 01:06:22,930 This is not-- it's the principle normal vectors. 1275 01:06:22,930 --> 01:06:24,726 There are many normals. 1276 01:06:24,726 --> 01:06:26,630 There is only one tangent direction, 1277 01:06:26,630 --> 01:06:29,010 but in terms of normals there are many that 1278 01:06:29,010 --> 01:06:30,914 are-- all of these are normals. 1279 01:06:30,914 --> 01:06:34,940 All the perpendicular in the plane-- [INAUDIBLE] 1280 01:06:34,940 --> 01:06:39,780 so this is my law of motion, T. All this plane is normal. 1281 01:06:39,780 --> 01:06:41,960 So any of these vectors is a normal. 1282 01:06:41,960 --> 01:06:44,990 The one we choose and defined as T prime 1283 01:06:44,990 --> 01:06:47,220 over T prime [INAUDIBLE] absolute values 1284 01:06:47,220 --> 01:06:48,990 called the principal normal. 1285 01:06:48,990 --> 01:06:51,350 It's like the principal of a high school. 1286 01:06:51,350 --> 01:06:53,230 He is important. 1287 01:06:53,230 --> 01:06:58,352 So T and B-- B goes down, or goes-- down. 1288 01:06:58,352 --> 01:07:04,540 Well, yeah, because B is T cross N. So when you find the Frenet 1289 01:07:04,540 --> 01:07:10,440 Trihedron, TNB, it's like that. 1290 01:07:10,440 --> 01:07:15,685 T, N, and B. What's special, why do we call it the frame, 1291 01:07:15,685 --> 01:07:18,460 is that every [? payer ?] of vectors 1292 01:07:18,460 --> 01:07:20,090 are mutually orthogonal. 1293 01:07:20,090 --> 01:07:22,270 And they are all unit vectors. 1294 01:07:22,270 --> 01:07:25,910 This is the famous Frenet frame. 1295 01:07:25,910 --> 01:07:27,580 Now, Mr. Frenet was a smart guy. 1296 01:07:27,580 --> 01:07:32,330 He found-- I don't know whether he was adopting mathematics 1297 01:07:32,330 --> 01:07:33,050 or not. 1298 01:07:33,050 --> 01:07:34,290 Doesn't matter. 1299 01:07:34,290 --> 01:07:37,970 He found a bunch of formulas, of which this is the first one. 1300 01:07:37,970 --> 01:07:42,265 And it's called a first Frenet formula. 1301 01:07:42,265 --> 01:07:44,230 That's one thing I want to ask you. 1302 01:07:44,230 --> 01:07:47,170 And then I'm going to give you more formulas for curvatures, 1303 01:07:47,170 --> 01:07:50,460 depending on how you define your curve. 1304 01:07:50,460 --> 01:08:08,832 So for example, base B based on the definition one 1305 01:08:08,832 --> 01:08:18,870 can prove that for a curve that is not parametrizing 1306 01:08:18,870 --> 01:08:22,870 arclength-- you say, ugh, forget about parametrization 1307 01:08:22,870 --> 01:08:23,590 in arclength. 1308 01:08:23,590 --> 01:08:26,840 This time you're assuming, I want to know! 1309 01:08:26,840 --> 01:08:29,410 I'm coming to this because Ryan asked. 1310 01:08:29,410 --> 01:08:32,381 I want to know, what is the formula directly? 1311 01:08:32,381 --> 01:08:34,439 Is there a direct formula that comes 1312 01:08:34,439 --> 01:08:38,529 from here for the curvature? 1313 01:08:38,529 --> 01:08:41,310 Yeah, but it's a lot more complicated. 1314 01:08:41,310 --> 01:08:45,265 When I was a freshman, maybe a freshman or a sophomore, 1315 01:08:45,265 --> 01:08:48,090 I don't remember, when I was asked to memorize 1316 01:08:48,090 --> 01:08:52,689 that, I did not memorize it. 1317 01:08:52,689 --> 01:08:56,649 Then when I started working as a faculty member, 1318 01:08:56,649 --> 01:09:01,810 I saw that I am supposed to ask it from my students. 1319 01:09:01,810 --> 01:09:05,578 So this is going to be R prime plus product 1320 01:09:05,578 --> 01:09:12,057 R double prime in magnitude over R prime cubed. 1321 01:09:12,057 --> 01:09:14,550 So how am I supposed to remember that? 1322 01:09:14,550 --> 01:09:15,658 It's not so easy. 1323 01:09:15,658 --> 01:09:17,800 Are you cold there? 1324 01:09:17,800 --> 01:09:18,649 It's cold there. 1325 01:09:18,649 --> 01:09:22,590 I don't know how these roofs are made. 1326 01:09:22,590 --> 01:09:24,670 Velocity times acceleration. 1327 01:09:24,670 --> 01:09:26,620 This is what I try to teach myself. 1328 01:09:26,620 --> 01:09:29,810 I was old already, 26 or 27. 1329 01:09:29,810 --> 01:09:32,979 Velocity times acceleration, cross product, 1330 01:09:32,979 --> 01:09:35,760 take the magnitude, divide by the speed, cube. 1331 01:09:35,760 --> 01:09:36,810 Oh my god. 1332 01:09:36,810 --> 01:09:41,340 So I was supposed to know that when I was 18 or 19. 1333 01:09:41,340 --> 01:09:44,510 Now, I was teaching majors of mechanical engineering. 1334 01:09:44,510 --> 01:09:45,840 They knew that by heart. 1335 01:09:45,840 --> 01:09:48,210 I didn't, so I had to learn it. 1336 01:09:48,210 --> 01:09:51,475 So if one is too lazy or it's simply 1337 01:09:51,475 --> 01:09:54,915 inconvenient to try to reparametrize from R of T 1338 01:09:54,915 --> 01:10:00,790 being arclength parametrization R of s and do that thing here, 1339 01:10:00,790 --> 01:10:05,300 one can just plug in and find the curvature like that. 1340 01:10:05,300 --> 01:10:08,450 For example, guys, as Ryan asked, 1341 01:10:08,450 --> 01:10:13,290 if I have A cosine, [INAUDIBLE], and I do this with respect 1342 01:10:13,290 --> 01:10:16,740 to T, can I get k without-- k will not 1343 01:10:16,740 --> 01:10:18,950 depend on T or s or tau. 1344 01:10:18,950 --> 01:10:20,860 It will always be the same. 1345 01:10:20,860 --> 01:10:23,400 I will still get A over A squared plus B 1346 01:10:23,400 --> 01:10:25,260 squared, no matter what. 1347 01:10:25,260 --> 01:10:28,930 So even if I use this formula for my helix, 1348 01:10:28,930 --> 01:10:30,950 I'm going to get the same thing. 1349 01:10:30,950 --> 01:10:33,060 I'll get A over A squared plus B squared, 1350 01:10:33,060 --> 01:10:35,390 because curvature is an invariant. 1351 01:10:35,390 --> 01:10:38,510 There is another invariant that's-- the other invariant, 1352 01:10:38,510 --> 01:10:40,550 of course, in space is called torsion. 1353 01:10:40,550 --> 01:10:43,680 We want to talk a little bit about that later. 1354 01:10:43,680 --> 01:10:48,780 So is this hard? 1355 01:10:48,780 --> 01:10:49,280 No. 1356 01:10:49,280 --> 01:10:50,450 It shouldn't be hard. 1357 01:10:50,450 --> 01:10:54,930 And you guys should be able to help me on that, hopefully. 1358 01:10:54,930 --> 01:10:56,600 How do we prove that? 1359 01:10:56,600 --> 01:10:58,480 STUDENT: N is G prime [INAUDIBLE]. 1360 01:10:58,480 --> 01:11:01,950 1361 01:11:01,950 --> 01:11:03,450 PROFESSOR TODA: That's right, proof. 1362 01:11:03,450 --> 01:11:06,080 And that's a very good start, wouldn't you say? 1363 01:11:06,080 --> 01:11:09,090 So what were the definitions? 1364 01:11:09,090 --> 01:11:14,350 Let me start from the definition of T. 1365 01:11:14,350 --> 01:11:17,180 That's going to be-- I am in hard planes, right? 1366 01:11:17,180 --> 01:11:21,050 So you say, wait, why do you write it as a quotient? 1367 01:11:21,050 --> 01:11:22,430 You're being silly. 1368 01:11:22,430 --> 01:11:24,530 You are in arclength, Magdalena. 1369 01:11:24,530 --> 01:11:25,470 I am. 1370 01:11:25,470 --> 01:11:26,340 I am. 1371 01:11:26,340 --> 01:11:29,860 I just pretend that I cannot see that. 1372 01:11:29,860 --> 01:11:32,160 So if I'm in arclength, that means 1373 01:11:32,160 --> 01:11:35,870 that the denominator is 1. 1374 01:11:35,870 --> 01:11:37,320 So I'm being silly. 1375 01:11:37,320 --> 01:11:44,380 So R prime of s is T. Say it again. 1376 01:11:44,380 --> 01:11:49,280 R prime of s is T. OK. 1377 01:11:49,280 --> 01:11:53,720 Now, did we know that T and N are orthogonal? 1378 01:11:53,720 --> 01:12:00,530 1379 01:12:00,530 --> 01:12:04,350 How did we know that T and N were orthogonal? 1380 01:12:04,350 --> 01:12:07,524 We proved that last time, actually. 1381 01:12:07,524 --> 01:12:11,003 T and N are orthogonal. 1382 01:12:11,003 --> 01:12:12,991 How do I write that? [INAUDIBLE]. 1383 01:12:12,991 --> 01:12:15,973 1384 01:12:15,973 --> 01:12:21,990 Meaning that T is perpendicular to N, right? 1385 01:12:21,990 --> 01:12:24,110 From the definition. 1386 01:12:24,110 --> 01:12:26,000 You said it right, Sandra. 1387 01:12:26,000 --> 01:12:28,000 But why is it from the definition 1388 01:12:28,000 --> 01:12:30,900 that I can jump to conclusions and say, oh, 1389 01:12:30,900 --> 01:12:36,120 since I have T prime here, then this is perpendicular to T? 1390 01:12:36,120 --> 01:12:37,435 Well, we did that last time. 1391 01:12:37,435 --> 01:12:39,236 STUDENT: Two parallel vectors. 1392 01:12:39,236 --> 01:12:41,110 PROFESSOR TODA: We did it-- how did we do it? 1393 01:12:41,110 --> 01:12:42,250 We did this last. 1394 01:12:42,250 --> 01:12:45,800 We said T dot T equals 1. 1395 01:12:45,800 --> 01:12:47,960 Prime the whole thing. 1396 01:12:47,960 --> 01:12:54,270 T prime times T plus T times T prime, T dot T prime will be 0. 1397 01:12:54,270 --> 01:12:57,030 So T and T prime are perpendicular always. 1398 01:12:57,030 --> 01:12:58,150 Right? 1399 01:12:58,150 --> 01:13:03,030 OK, so the whole thing is a colinear vector to T prime. 1400 01:13:03,030 --> 01:13:05,025 It's just T prime times the scalar. 1401 01:13:05,025 --> 01:13:08,320 So he must be perpendicular to T. 1402 01:13:08,320 --> 01:13:10,720 So T and N are perpendicular. 1403 01:13:10,720 --> 01:13:14,680 So I do have the direction of motion. 1404 01:13:14,680 --> 01:13:19,482 I know that I must have some scalar here. 1405 01:13:19,482 --> 01:13:22,796 1406 01:13:22,796 --> 01:13:26,630 How do I prove that this scalar is the curvature? 1407 01:13:26,630 --> 01:13:30,510 1408 01:13:30,510 --> 01:13:35,995 So if I have-- if they are colinear-- why are 1409 01:13:35,995 --> 01:13:36,740 they colinear? 1410 01:13:36,740 --> 01:13:42,200 T perpendicular to T prime implies that T prime 1411 01:13:42,200 --> 01:13:46,020 is colinear to N. Say it again. 1412 01:13:46,020 --> 01:13:49,860 If T and T prime are perpendicular to one another, 1413 01:13:49,860 --> 01:13:53,260 that means T prime is calling it to the normal. 1414 01:13:53,260 --> 01:13:58,471 So here I may have alph-- no alpha. 1415 01:13:58,471 --> 01:14:00,260 I don't know! 1416 01:14:00,260 --> 01:14:03,700 Alpha over [INAUDIBLE] sounds like a curve. 1417 01:14:03,700 --> 01:14:04,616 Give me some function. 1418 01:14:04,616 --> 01:14:08,864 1419 01:14:08,864 --> 01:14:09,530 STUDENT: u of s? 1420 01:14:09,530 --> 01:14:11,050 PROFESSOR TODA: Gamma of s. 1421 01:14:11,050 --> 01:14:15,360 u of s, I don't know. 1422 01:14:15,360 --> 01:14:17,410 So how did I conclude that? 1423 01:14:17,410 --> 01:14:19,500 From T perpendicular to T prime. 1424 01:14:19,500 --> 01:14:22,210 Now from here on, you have to tell me why 1425 01:14:22,210 --> 01:14:29,430 gamma must be exactly kappa. 1426 01:14:29,430 --> 01:14:33,712 Well, let's take T prime from here. 1427 01:14:33,712 --> 01:14:38,090 T prime from here will give me what? 1428 01:14:38,090 --> 01:14:40,730 T prime is our prime prime. 1429 01:14:40,730 --> 01:14:42,300 Say what? 1430 01:14:42,300 --> 01:14:43,200 Our prime prime. 1431 01:14:43,200 --> 01:14:44,657 What is our prime prime? 1432 01:14:44,657 --> 01:14:46,565 Our [? problem ?] prime of s. 1433 01:14:46,565 --> 01:14:48,582 STUDENT: You have one too many primes inside. 1434 01:14:48,582 --> 01:14:49,665 PROFESSOR TODA: Oh my god. 1435 01:14:49,665 --> 01:14:50,165 Yeah. 1436 01:14:50,165 --> 01:14:52,770 1437 01:14:52,770 --> 01:14:54,120 So R prime prime. 1438 01:14:54,120 --> 01:14:58,430 So T prime in absolute value will 1439 01:14:58,430 --> 01:15:02,650 be exactly R double prime of s. 1440 01:15:02,650 --> 01:15:04,990 Oh, OK. 1441 01:15:04,990 --> 01:15:10,130 Note that from here also T prime of s in absolute value, 1442 01:15:10,130 --> 01:15:13,840 in magnitude, I'm sorry, has to be gamma of s. 1443 01:15:13,840 --> 01:15:14,820 Why is that? 1444 01:15:14,820 --> 01:15:17,470 Because the magnitude of N is 1. 1445 01:15:17,470 --> 01:15:20,930 N is unique vector by definition. 1446 01:15:20,930 --> 01:15:24,870 So these two guys have to coincide. 1447 01:15:24,870 --> 01:15:27,132 So R double prime, the best thing 1448 01:15:27,132 --> 01:15:28,590 that I need to do, it must coincide 1449 01:15:28,590 --> 01:15:30,500 with the scalar gamma of s. 1450 01:15:30,500 --> 01:15:32,840 So who is the mysterious gamma of s? 1451 01:15:32,840 --> 01:15:36,340 He has no chance but being this guy. 1452 01:15:36,340 --> 01:15:38,660 But this guy has a name. 1453 01:15:38,660 --> 01:15:41,920 This guy, he's the curvature [? cap ?] of s by definition. 1454 01:15:41,920 --> 01:15:45,738 1455 01:15:45,738 --> 01:15:49,126 Remember, Ryan, this is the definition. 1456 01:15:49,126 --> 01:15:51,546 So by definition the curvature was the magnitude 1457 01:15:51,546 --> 01:15:55,010 of the acceleration in arclength. 1458 01:15:55,010 --> 01:15:55,910 OK. 1459 01:15:55,910 --> 01:15:58,460 Both of these guys are T prime in magnitude. 1460 01:15:58,460 --> 01:16:01,770 So they must be equal from here and here. 1461 01:16:01,770 --> 01:16:04,696 It implies that my gamma must be kappa. 1462 01:16:04,696 --> 01:16:07,780 And I prove the formula. 1463 01:16:07,780 --> 01:16:09,130 OK. 1464 01:16:09,130 --> 01:16:10,911 How do you say something is proved? 1465 01:16:10,911 --> 01:16:12,294 Because this is what we wanted. 1466 01:16:12,294 --> 01:16:16,235 We wanted to replace this generic scalar function 1467 01:16:16,235 --> 01:16:20,080 to prove that this is just the curvature. 1468 01:16:20,080 --> 01:16:20,580 QED. 1469 01:16:20,580 --> 01:16:24,420 1470 01:16:24,420 --> 01:16:26,960 That's exactly what we wanted to prove. 1471 01:16:26,960 --> 01:16:29,046 Now, whatever scalar function you have here, 1472 01:16:29,046 --> 01:16:30,170 that must be the curvature. 1473 01:16:30,170 --> 01:16:34,460 1474 01:16:34,460 --> 01:16:36,415 Very smart guy, this Mr. Frenet. 1475 01:16:36,415 --> 01:16:39,720 1476 01:16:39,720 --> 01:16:40,990 I'm now going to take a break. 1477 01:16:40,990 --> 01:16:43,830 If you want to go use the bathroom really quickly, 1478 01:16:43,830 --> 01:16:45,025 feel free to do it. 1479 01:16:45,025 --> 01:16:47,590 1480 01:16:47,590 --> 01:16:49,470 I'm just going to clean the board, 1481 01:16:49,470 --> 01:16:51,694 and I'll keep going in a few minutes. 1482 01:16:51,694 --> 01:17:50,501 1483 01:17:50,501 --> 01:17:51,334 STUDENT: [INAUDIBLE] 1484 01:17:51,334 --> 01:17:55,799 1485 01:17:55,799 --> 01:17:57,298 PROFESSOR TODA: I will do it-- well, 1486 01:17:57,298 --> 01:18:01,274 actually I want to do a different example, simple one, 1487 01:18:01,274 --> 01:18:06,244 which is a plain curve, and show that the curvature has a very 1488 01:18:06,244 --> 01:18:11,036 pretty formula that you could [INAUDIBLE] memorize, 1489 01:18:11,036 --> 01:18:13,516 that in essence is the same. 1490 01:18:13,516 --> 01:18:17,484 But it depends on y equals f of x. 1491 01:18:17,484 --> 01:18:19,468 [INAUDIBLE] So if somebody gives you 1492 01:18:19,468 --> 01:18:22,444 a plane called y equals f of x, can you 1493 01:18:22,444 --> 01:18:25,420 write that curvature [INAUDIBLE] function of f? 1494 01:18:25,420 --> 01:18:26,908 And you can. 1495 01:18:26,908 --> 01:18:30,545 And again, I was deep in that when I was 18 or 19 1496 01:18:30,545 --> 01:18:31,868 as a freshman. 1497 01:18:31,868 --> 01:18:35,836 But unfortunately for me I didn't learn it at that time. 1498 01:18:35,836 --> 01:18:41,340 And several years later when I started teaching engineers, 1499 01:18:41,340 --> 01:18:43,776 well, they are mostly mechanical. 1500 01:18:43,776 --> 01:18:46,770 And mechanical engineering [INAUDIBLE]. 1501 01:18:46,770 --> 01:18:50,263 They knew those, and they needed those in every research paper. 1502 01:18:50,263 --> 01:18:54,255 So I had to learn it together with them. 1503 01:18:54,255 --> 01:18:57,748 I'll worry about [INAUDIBLE]. 1504 01:18:57,748 --> 01:19:01,241 STUDENT: Can you do a really ugly one, like [INAUDIBLE]? 1505 01:19:01,241 --> 01:19:04,734 PROFESSOR TODA: I can do some ugly ones. 1506 01:19:04,734 --> 01:20:37,240 1507 01:20:37,240 --> 01:20:47,670 And once you know the general parametrization, 1508 01:20:47,670 --> 01:20:51,908 it will give you a curvature. 1509 01:20:51,908 --> 01:20:53,327 Now I'm testing your memory. 1510 01:20:53,327 --> 01:20:55,230 Let's see what you remember. 1511 01:20:55,230 --> 01:20:59,854 Um-- don't look at the notes. 1512 01:20:59,854 --> 01:21:03,270 A positive function, absolute-- actually, 1513 01:21:03,270 --> 01:21:06,680 magnitude of what vector? 1514 01:21:06,680 --> 01:21:07,580 STUDENT: R prime. 1515 01:21:07,580 --> 01:21:17,060 PROFESSOR TODA: R prime velocity plus acceleration speed cubed. 1516 01:21:17,060 --> 01:21:18,540 Right? 1517 01:21:18,540 --> 01:21:19,040 OK. 1518 01:21:19,040 --> 01:21:24,070 Now, can we take advantage of what we just learned 1519 01:21:24,070 --> 01:21:30,055 and find-- you find with me, of course, not 1520 01:21:30,055 --> 01:21:33,654 as professor and student, but like a group of students 1521 01:21:33,654 --> 01:21:35,070 together. 1522 01:21:35,070 --> 01:21:46,286 Let's find a simple formula corresponding 1523 01:21:46,286 --> 01:21:52,202 to the curvature of a plane curve. 1524 01:21:52,202 --> 01:21:59,597 1525 01:21:59,597 --> 01:22:05,280 And the plane curve could be [INAUDIBLE] 1526 01:22:05,280 --> 01:22:09,020 in two different ways, just because I want 1527 01:22:09,020 --> 01:22:14,510 you to practice more on that. 1528 01:22:14,510 --> 01:22:18,360 Either given as a general parametrization-- guys, 1529 01:22:18,360 --> 01:22:20,090 what is the general parametrization 1530 01:22:20,090 --> 01:22:24,510 I'm talking about for a plane curve? 1531 01:22:24,510 --> 01:22:26,400 x of t, y of t, right? 1532 01:22:26,400 --> 01:22:28,660 x equals x of t. 1533 01:22:28,660 --> 01:22:29,870 y equals y of t. 1534 01:22:29,870 --> 01:22:34,400 So one should not have to do that all the time, 1535 01:22:34,400 --> 01:22:37,310 not have to do that for a simplification like a playing 1536 01:22:37,310 --> 01:22:38,190 card. 1537 01:22:38,190 --> 01:22:41,710 We have to find another formula that's pretty, right? 1538 01:22:41,710 --> 01:22:43,210 Well, maybe it's not as pretty. 1539 01:22:43,210 --> 01:22:45,250 But when is it really pretty? 1540 01:22:45,250 --> 01:22:49,090 I bet it's going to be really pretty if you have a plane 1541 01:22:49,090 --> 01:22:54,610 curve even as you're used to in an explicit form-- 1542 01:22:54,610 --> 01:22:56,600 I keep going. 1543 01:22:56,600 --> 01:22:59,910 No stop. [INAUDIBLE]. 1544 01:22:59,910 --> 01:23:01,110 I think it's better. 1545 01:23:01,110 --> 01:23:03,450 We make better use of time this way. 1546 01:23:03,450 --> 01:23:06,890 Or y equals f of x. 1547 01:23:06,890 --> 01:23:12,600 1548 01:23:12,600 --> 01:23:17,953 This is an explicit way to write the equation of a curve. 1549 01:23:17,953 --> 01:23:20,660 1550 01:23:20,660 --> 01:23:23,330 OK, so what do we need to do? 1551 01:23:23,330 --> 01:23:26,430 That should be really easy. 1552 01:23:26,430 --> 01:23:32,958 R of t being the first case of our general parametrization, 1553 01:23:32,958 --> 01:23:40,720 x equals x of t, y equals y of t will be-- who tells me, guys, 1554 01:23:40,720 --> 01:23:43,470 that-- this is in your hands. 1555 01:23:43,470 --> 01:23:47,700 Now you convinced me that, for whatever reason, 1556 01:23:47,700 --> 01:23:49,880 you [INAUDIBLE]. 1557 01:23:49,880 --> 01:23:51,800 You became friends with these curves. 1558 01:23:51,800 --> 01:23:52,760 I don't know when. 1559 01:23:52,760 --> 01:23:54,680 I guess in the process of doing homework. 1560 01:23:54,680 --> 01:23:55,650 Am I right? 1561 01:23:55,650 --> 01:24:00,480 I think you did not quite like them before or the last week. 1562 01:24:00,480 --> 01:24:02,470 But I think you're friends with them now. 1563 01:24:02,470 --> 01:24:06,556 x of t, y of t. 1564 01:24:06,556 --> 01:24:07,990 Let people talk. 1565 01:24:07,990 --> 01:24:12,770 1566 01:24:12,770 --> 01:24:13,740 STUDENT: 0. 1567 01:24:13,740 --> 01:24:15,670 PROFESSOR TODA: So. 1568 01:24:15,670 --> 01:24:16,170 Great. 1569 01:24:16,170 --> 01:24:21,110 And then R prime of t will be x prime of t, y prime of t, 1570 01:24:21,110 --> 01:24:21,760 and 0. 1571 01:24:21,760 --> 01:24:24,487 I assume this to be always non-zero. 1572 01:24:24,487 --> 01:24:26,235 I have a regular curve. 1573 01:24:26,235 --> 01:24:30,680 R double prime will be-- x double prime where 1574 01:24:30,680 --> 01:24:34,146 double prime-- we did the review today 1575 01:24:34,146 --> 01:24:36,450 of the lasting acceleration. 1576 01:24:36,450 --> 01:24:39,630 Now, your friends over here, are they nice or mean? 1577 01:24:39,630 --> 01:24:42,600 I hope they are not so mean. 1578 01:24:42,600 --> 01:24:45,810 The cross product is a friendly fellow. 1579 01:24:45,810 --> 01:24:48,970 You have i, j, k, and then the second row 1580 01:24:48,970 --> 01:24:50,640 would be x prime, y prime, 0. 1581 01:24:50,640 --> 01:24:54,846 The last row would be x double prime, y double prime, 0. 1582 01:24:54,846 --> 01:24:58,550 And it's a piece of cake. 1583 01:24:58,550 --> 01:25:02,146 1584 01:25:02,146 --> 01:25:03,520 OK, piece of cake, piece of cake. 1585 01:25:03,520 --> 01:25:08,960 But I want to know what the answer is. 1586 01:25:08,960 --> 01:25:15,630 So you have exactly 15 seconds to answer this question. 1587 01:25:15,630 --> 01:25:23,298 Who is R prime plus R double prime as a [? coordinate. ?] 1588 01:25:23,298 --> 01:25:25,266 [INTERPOSING VOICES] 1589 01:25:25,266 --> 01:25:28,710 1590 01:25:28,710 --> 01:25:29,700 PROFESSOR TODA: Good. 1591 01:25:29,700 --> 01:25:35,372 x prime, y double prime minus x double prime, y prime times k. 1592 01:25:35,372 --> 01:25:37,720 And it doesn't matter when I take the magnitude, 1593 01:25:37,720 --> 01:25:40,600 because magnitude of k is 1. 1594 01:25:40,600 --> 01:25:42,380 So I discovered some. 1595 01:25:42,380 --> 01:25:46,700 This is how mathematicians like to discover new formulas based 1596 01:25:46,700 --> 01:25:48,730 on the formulas they [? knew. ?] They 1597 01:25:48,730 --> 01:25:50,090 have a lot of satisfaction. 1598 01:25:50,090 --> 01:25:51,020 Look what I got. 1599 01:25:51,020 --> 01:25:56,630 Of course, they in general have more complicated things to do, 1600 01:25:56,630 --> 01:25:58,952 and they have to check and recheck. 1601 01:25:58,952 --> 01:26:06,305 But every piece of a computation is a challenge. 1602 01:26:06,305 --> 01:26:10,310 And that gives people satisfaction. 1603 01:26:10,310 --> 01:26:14,900 And when they make a mistake, it brings a lot of tears as well. 1604 01:26:14,900 --> 01:26:21,190 So what-- could be written on the bottom, what's 1605 01:26:21,190 --> 01:26:24,680 the speed cubed? 1606 01:26:24,680 --> 01:26:26,740 Speed is coming from this guy. 1607 01:26:26,740 --> 01:26:32,055 So the speed of the velocity, the magnitude of the velocity 1608 01:26:32,055 --> 01:26:32,990 is the speed. 1609 01:26:32,990 --> 01:26:35,100 And that-- going to give you square. 1610 01:26:35,100 --> 01:26:37,010 I'm not going to write down [INAUDIBLE]. 1611 01:26:37,010 --> 01:26:39,144 Square root of x squared, x prime squared times 1612 01:26:39,144 --> 01:26:42,740 y prime squared, and I cube that. 1613 01:26:42,740 --> 01:26:46,370 Many people, and I saw that in engineering, they 1614 01:26:46,370 --> 01:26:49,760 don't like to put that square root anymore. 1615 01:26:49,760 --> 01:26:53,950 And they just write x prime squared plus y prime squared 1616 01:26:53,950 --> 01:26:55,110 to the what power? 1617 01:26:55,110 --> 01:26:55,690 STUDENT: 3/2. 1618 01:26:55,690 --> 01:26:56,540 PROFESSOR TODA: 3/2. 1619 01:26:56,540 --> 01:27:01,650 So this is very useful for engineering styles, 1620 01:27:01,650 --> 01:27:05,110 when you have to deal with plane curves, motions 1621 01:27:05,110 --> 01:27:08,560 in plane curves. 1622 01:27:08,560 --> 01:27:13,770 But now what do you have in the case, 1623 01:27:13,770 --> 01:27:19,220 in the happy case, when you have y equals f of x? 1624 01:27:19,220 --> 01:27:21,335 I'm going to do that in a second. 1625 01:27:21,335 --> 01:27:26,460 1626 01:27:26,460 --> 01:27:29,144 I want to keep this formula on the board. 1627 01:27:29,144 --> 01:27:38,108 1628 01:27:38,108 --> 01:27:40,598 What's the simplest parametrization? 1629 01:27:40,598 --> 01:27:43,088 Because that's why we need it, to look over 1630 01:27:43,088 --> 01:27:46,580 parametrizations again and again. 1631 01:27:46,580 --> 01:27:52,190 R of t for this plane curve will be-- what is t? 1632 01:27:52,190 --> 01:27:53,750 x is t, right? 1633 01:27:53,750 --> 01:27:56,000 x is t, y is f of t. 1634 01:27:56,000 --> 01:27:57,050 Piece of cake. 1635 01:27:57,050 --> 01:28:00,310 So you have t and f of t. 1636 01:28:00,310 --> 01:28:03,355 And how many of you watched the videos that I sent you? 1637 01:28:03,355 --> 01:28:06,394 1638 01:28:06,394 --> 01:28:08,890 Do you prefer Khan Academy, or do you 1639 01:28:08,890 --> 01:28:12,880 prefer the guys, [INAUDIBLE] guys who are lecturing? 1640 01:28:12,880 --> 01:28:15,585 The professors who are lecturing in front of a board or in front 1641 01:28:15,585 --> 01:28:17,955 of a-- what is that? 1642 01:28:17,955 --> 01:28:21,350 A projector screen? 1643 01:28:21,350 --> 01:28:22,620 I like all of them. 1644 01:28:22,620 --> 01:28:25,140 I think they're very good. 1645 01:28:25,140 --> 01:28:27,572 I think you can learn a lot from three 1646 01:28:27,572 --> 01:28:29,530 or four different instructors at the same time. 1647 01:28:29,530 --> 01:28:32,010 That's ideal. 1648 01:28:32,010 --> 01:28:35,590 I guess that you have this chance only now 1649 01:28:35,590 --> 01:28:36,980 in the past few years. 1650 01:28:36,980 --> 01:28:41,300 Because 20 years ago, if you're didn't like your instructor 1651 01:28:41,300 --> 01:28:45,790 or just you couldn't stand them, you had no other chance. 1652 01:28:45,790 --> 01:28:48,030 There was no YouTube, no internet, 1653 01:28:48,030 --> 01:28:50,972 no way to learn from others. 1654 01:28:50,972 --> 01:29:00,220 R prime of t would be 1 f prime of t. 1655 01:29:00,220 --> 01:29:02,840 But instead of t I'll out x, because x is t. 1656 01:29:02,840 --> 01:29:03,930 I don't care. 1657 01:29:03,930 --> 01:29:07,470 R double prime of t would be 0, f double prime of x. 1658 01:29:07,470 --> 01:29:12,430 So I feel that, hey, I know what's going to come up. 1659 01:29:12,430 --> 01:29:15,390 And I'm ready. 1660 01:29:15,390 --> 01:29:17,680 Well, we are ready to write it down. 1661 01:29:17,680 --> 01:29:20,190 This is going to be Mr. x prime. 1662 01:29:20,190 --> 01:29:22,790 This is going to be replacing Mr. y prime. 1663 01:29:22,790 --> 01:29:25,780 This is going to replace Mr. a double prime. 1664 01:29:25,780 --> 01:29:29,350 This is going to be replacing Mr. y double prime of x. 1665 01:29:29,350 --> 01:29:31,120 Oh, OK, all right. 1666 01:29:31,120 --> 01:29:38,840 So k, our old friend from here will become what? 1667 01:29:38,840 --> 01:29:42,490 And I'd better shut up, because I'm talking too much. 1668 01:29:42,490 --> 01:29:45,020 STUDENT: [INAUDIBLE] double prime [INAUDIBLE]. 1669 01:29:45,020 --> 01:29:48,092 PROFESSOR TODA: That is the absolute value, mm-hmm. 1670 01:29:48,092 --> 01:29:54,041 [? n ?] double prime of x, and nothing else. 1671 01:29:54,041 --> 01:29:54,540 Right, guys? 1672 01:29:54,540 --> 01:29:55,628 Are you with me? 1673 01:29:55,628 --> 01:29:56,960 Divided by-- 1674 01:29:56,960 --> 01:29:57,850 STUDENT: [INAUDIBLE] 1675 01:29:57,850 --> 01:29:59,558 PROFESSOR TODA: Should I add square root? 1676 01:29:59,558 --> 01:30:00,890 I love square roots. 1677 01:30:00,890 --> 01:30:01,880 I'm crazy about them. 1678 01:30:01,880 --> 01:30:11,950 So you go 1 plus f prime squared cubed. 1679 01:30:11,950 --> 01:30:16,410 So that's going to be-- any questions? 1680 01:30:16,410 --> 01:30:18,090 Are you guys with me? 1681 01:30:18,090 --> 01:30:21,296 That's going to be the formula that I'm going 1682 01:30:21,296 --> 01:30:22,420 to use in the next example. 1683 01:30:22,420 --> 01:30:25,540 1684 01:30:25,540 --> 01:30:29,820 In case somebody wants to know-- I got 1685 01:30:29,820 --> 01:30:32,010 this question from one of you. 1686 01:30:32,010 --> 01:30:35,060 Suppose we get a parametrization of a circle 1687 01:30:35,060 --> 01:30:37,410 in the midterm or the final. 1688 01:30:37,410 --> 01:30:44,335 Somebody says, I have x of t, just like we did it 1689 01:30:44,335 --> 01:30:47,555 today, a cosine t plus 0. 1690 01:30:47,555 --> 01:30:52,380 And y of t equals a sine t plus y 0. 1691 01:30:52,380 --> 01:30:53,990 What is this, guys? 1692 01:30:53,990 --> 01:31:04,495 This is a circle, a center at 0, y, 0, and radius a. 1693 01:31:04,495 --> 01:31:08,630 1694 01:31:08,630 --> 01:31:14,916 Can use a better formula-- that anticipated my action today-- 1695 01:31:14,916 --> 01:31:18,535 to actually prove that k is going to be [? 1/a? ?] 1696 01:31:18,535 --> 01:31:19,260 Precisely. 1697 01:31:19,260 --> 01:31:20,550 Can we do that in the exam? 1698 01:31:20,550 --> 01:31:23,120 Yes. 1699 01:31:23,120 --> 01:31:25,490 So while I told you a long time ago 1700 01:31:25,490 --> 01:31:28,590 that engineers and mathematicians observed 1701 01:31:28,590 --> 01:31:31,000 hundreds of years ago-- actually, 1702 01:31:31,000 --> 01:31:32,770 somebody said, no, you're not right. 1703 01:31:32,770 --> 01:31:35,010 The Egyptians already saw that. 1704 01:31:35,010 --> 01:31:38,243 They had the notion of inverse proportionality 1705 01:31:38,243 --> 01:31:42,390 in Egypt, which makes sense if you look at the pyramids. 1706 01:31:42,390 --> 01:31:47,750 So one look at the radius, it says if the radius is 2, 1707 01:31:47,750 --> 01:31:50,640 then the curvature is not very bent. 1708 01:31:50,640 --> 01:31:52,760 So the curvature's inverse proportion [INAUDIBLE] 1709 01:31:52,760 --> 01:31:53,480 the radius. 1710 01:31:53,480 --> 01:31:57,480 So if this is 2, we said the curvature's 1/2. 1711 01:31:57,480 --> 01:32:01,680 If you take a big circle, the bigger 1712 01:32:01,680 --> 01:32:04,060 the radius, the smaller the bending 1713 01:32:04,060 --> 01:32:07,640 of the arc of the circle, the smaller of the curvature. 1714 01:32:07,640 --> 01:32:10,670 Apparently the ancient world knew that already. 1715 01:32:10,670 --> 01:32:12,195 They Egyptians knew that. 1716 01:32:12,195 --> 01:32:13,140 The Greeks knew that. 1717 01:32:13,140 --> 01:32:15,432 But I think they never formalized it-- 1718 01:32:15,432 --> 01:32:16,624 not that I know. 1719 01:32:16,624 --> 01:32:19,290 1720 01:32:19,290 --> 01:32:24,530 So if you are asked to do this in any exam, 1721 01:32:24,530 --> 01:32:26,942 do you think that would be a problem? 1722 01:32:26,942 --> 01:32:28,150 Of course we would do review. 1723 01:32:28,150 --> 01:32:31,730 Because people are going to forget this formula, or even 1724 01:32:31,730 --> 01:32:33,370 the definition. 1725 01:32:33,370 --> 01:32:36,010 You can compute k for this formula. 1726 01:32:36,010 --> 01:32:39,330 And we are going to get k to the 1/a. 1727 01:32:39,330 --> 01:32:41,340 This is a piece of cake, actually. 1728 01:32:41,340 --> 01:32:44,110 You may not believe me, but once you plug in the equations 1729 01:32:44,110 --> 01:32:46,290 it's very easy. 1730 01:32:46,290 --> 01:32:48,425 Or you can do it from the definition 1731 01:32:48,425 --> 01:32:50,640 that gives you k of s. 1732 01:32:50,640 --> 01:32:52,560 You'll reparametrize this in arclength. 1733 01:32:52,560 --> 01:32:54,710 You can do that as well. 1734 01:32:54,710 --> 01:32:57,600 And you still get 1/a. 1735 01:32:57,600 --> 01:32:59,880 The question that I got by email, 1736 01:32:59,880 --> 01:33:01,440 and I get a lot of email. 1737 01:33:01,440 --> 01:33:04,660 I told you, that keeps me busy a lot, 1738 01:33:04,660 --> 01:33:08,490 about 200 emails every day. 1739 01:33:08,490 --> 01:33:10,680 I really like the emails I get from students, 1740 01:33:10,680 --> 01:33:13,620 because I get emails from all sorts of sources-- 1741 01:33:13,620 --> 01:33:15,340 Got some spam also. 1742 01:33:15,340 --> 01:33:21,780 Anyway, what I'm trying to say, I got this question last time 1743 01:33:21,780 --> 01:33:24,680 saying, if on the midterm we get such a question, 1744 01:33:24,680 --> 01:33:28,073 can we say simply, curvature's 1/a, a is the radius. 1745 01:33:28,073 --> 01:33:30,820 Is that enough? 1746 01:33:30,820 --> 01:33:34,250 Depends on how the problem was formulated. 1747 01:33:34,250 --> 01:33:39,360 Most likely I'm going to make it through that or show that. 1748 01:33:39,360 --> 01:33:43,450 Even if you state something, like, yes, it's 1/a, 1749 01:33:43,450 --> 01:33:46,270 with a little argument, it's inverse proportional 1750 01:33:46,270 --> 01:33:50,380 to the radius, I will still give partial credit. 1751 01:33:50,380 --> 01:33:53,650 For any argument that is valid, especially 1752 01:33:53,650 --> 01:33:56,390 if it's based on empirical observation, 1753 01:33:56,390 --> 01:33:58,920 I do give some extra credit, even if you didn't 1754 01:33:58,920 --> 01:34:02,600 use the specific formula. 1755 01:34:02,600 --> 01:34:04,910 Let's see one example. 1756 01:34:04,910 --> 01:34:07,615 Let's take y equals e to the x. 1757 01:34:07,615 --> 01:34:11,870 1758 01:34:11,870 --> 01:34:15,726 No, let's take e to the negative x. 1759 01:34:15,726 --> 01:34:16,690 Doesn't matter. 1760 01:34:16,690 --> 01:34:20,560 1761 01:34:20,560 --> 01:34:26,375 y equals e to the negative x. 1762 01:34:26,375 --> 01:34:30,830 And let's make x between 0 and 1. 1763 01:34:30,830 --> 01:34:35,290 1764 01:34:35,290 --> 01:34:36,970 I'll say, write the curvature. 1765 01:34:36,970 --> 01:34:40,610 1766 01:34:40,610 --> 01:34:45,246 Write the equation or the formula of the curvature. 1767 01:34:45,246 --> 01:34:50,230 1768 01:34:50,230 --> 01:34:54,700 And I know it's 2 o'clock and I am answering questions. 1769 01:34:54,700 --> 01:34:58,070 This was a question that one of you had during the short break 1770 01:34:58,070 --> 01:34:59,300 we took. 1771 01:34:59,300 --> 01:35:00,480 Can we do such a problem? 1772 01:35:00,480 --> 01:35:01,544 Like she said. 1773 01:35:01,544 --> 01:35:02,960 Yes, I [INAUDIBLE] to the negative 1774 01:35:02,960 --> 01:35:05,520 x because I want to catch somebody 1775 01:35:05,520 --> 01:35:06,790 not knowing the derivative. 1776 01:35:06,790 --> 01:35:08,990 I don't know why I'm doing this. 1777 01:35:08,990 --> 01:35:10,510 Right? 1778 01:35:10,510 --> 01:35:15,480 So if I were to draw that, OK, try and draw that, but not now. 1779 01:35:15,480 --> 01:35:18,620 Now, what formula are you going to use? 1780 01:35:18,620 --> 01:35:21,860 Of course, you could do this in many ways. 1781 01:35:21,860 --> 01:35:24,510 All those formulas are equivalent for the curvature. 1782 01:35:24,510 --> 01:35:27,360 What's the simplest way to do it? 1783 01:35:27,360 --> 01:35:30,020 Do y prime. 1784 01:35:30,020 --> 01:35:33,770 Minus it to the minus x. 1785 01:35:33,770 --> 01:35:36,790 Note here in this problem that even if you mess up and forget 1786 01:35:36,790 --> 01:35:39,950 the minus sign, you still get the final answer correct. 1787 01:35:39,950 --> 01:35:46,050 But I may subtract a few points if I see something nonsensical. 1788 01:35:46,050 --> 01:35:47,300 y double prime equals-- 1789 01:35:47,300 --> 01:35:48,870 [INTERPOSING VOICES] 1790 01:35:48,870 --> 01:35:51,660 --plus e to the minus x. 1791 01:35:51,660 --> 01:35:56,738 And what is the curvature k of t? 1792 01:35:56,738 --> 01:35:59,200 STUDENT: y prime over-- 1793 01:35:59,200 --> 01:36:01,340 PROFESSOR TODA: Oh, I didn't say one more thing. 1794 01:36:01,340 --> 01:36:04,620 I want the curvature, but I also want the curvature 1795 01:36:04,620 --> 01:36:08,486 in three separate moments, in the beginning, in the end, 1796 01:36:08,486 --> 01:36:09,360 and in the middle. 1797 01:36:09,360 --> 01:36:11,430 STUDENT: Don't we need to parametrize it 1798 01:36:11,430 --> 01:36:15,250 so we can [INAUDIBLE] x prime [INAUDIBLE]? 1799 01:36:15,250 --> 01:36:16,740 PROFESSOR TODA: No. 1800 01:36:16,740 --> 01:36:18,340 Did I erase it? 1801 01:36:18,340 --> 01:36:19,330 STUDENT: Yeah, you did. 1802 01:36:19,330 --> 01:36:20,710 PROFESSOR TODA: [INAUDIBLE]. 1803 01:36:20,710 --> 01:36:24,356 And one of my colleagues said, Magda, you are smart, 1804 01:36:24,356 --> 01:36:28,000 but you are like one of those people who, 1805 01:36:28,000 --> 01:36:29,729 in the anecdotes about math professors, 1806 01:36:29,729 --> 01:36:31,520 gets out of their office and starts walking 1807 01:36:31,520 --> 01:36:32,930 and stops a student. 1808 01:36:32,930 --> 01:36:34,845 Was I going this way or that way? 1809 01:36:34,845 --> 01:36:36,070 And that's me. 1810 01:36:36,070 --> 01:36:37,650 And I'm sorry about that. 1811 01:36:37,650 --> 01:36:41,933 I should not have erased that. 1812 01:36:41,933 --> 01:36:44,000 I'm going to go ahead and rewrite it, 1813 01:36:44,000 --> 01:36:48,101 because I'm a goofball. 1814 01:36:48,101 --> 01:36:55,496 So the one that I wanted to use k of x will be f double prime. 1815 01:36:55,496 --> 01:36:56,975 STUDENT: And cubed. 1816 01:36:56,975 --> 01:36:58,454 PROFESSOR TODA: Cubed! 1817 01:36:58,454 --> 01:36:59,440 Thank you. 1818 01:36:59,440 --> 01:37:02,920 1819 01:37:02,920 --> 01:37:09,636 So that 3/2, remember it, [INAUDIBLE] 3/2 [INAUDIBLE] 1820 01:37:09,636 --> 01:37:10,990 square root cubed. 1821 01:37:10,990 --> 01:37:13,750 Now, for this one, is it hard? 1822 01:37:13,750 --> 01:37:15,080 No. 1823 01:37:15,080 --> 01:37:16,190 That's a piece of cake. 1824 01:37:16,190 --> 01:37:18,610 I said I like it in general, but I also 1825 01:37:18,610 --> 01:37:22,910 like it-- find the curvature of this curve in the beginning. 1826 01:37:22,910 --> 01:37:24,100 You travel on me. 1827 01:37:24,100 --> 01:37:27,590 From time 0 to 1 o'clock, whatever. 1828 01:37:27,590 --> 01:37:28,430 One second. 1829 01:37:28,430 --> 01:37:32,770 That's saying this is in seconds to make it more physical. 1830 01:37:32,770 --> 01:37:39,950 I want the k at 0, I want k at 1/2, and I want k at 1. 1831 01:37:39,950 --> 01:37:42,430 And I'd like you to compare those values. 1832 01:37:42,430 --> 01:37:46,270 1833 01:37:46,270 --> 01:37:49,120 And I'll give you one more task after that. 1834 01:37:49,120 --> 01:37:50,580 But let me start working. 1835 01:37:50,580 --> 01:37:53,090 So you say you help me on that. 1836 01:37:53,090 --> 01:37:55,100 [INAUDIBLE] 1837 01:37:55,100 --> 01:38:02,516 Minus x over square root of 1 plus-- 1838 01:38:02,516 --> 01:38:04,380 STUDENT: [INAUDIBLE] 1839 01:38:04,380 --> 01:38:05,312 PROFESSOR TODA: Right. 1840 01:38:05,312 --> 01:38:08,190 So can I write this differently, a little bit differently? 1841 01:38:08,190 --> 01:38:09,940 Like what? 1842 01:38:09,940 --> 01:38:12,020 I don't want to square each of the minus 2x. 1843 01:38:12,020 --> 01:38:14,350 Can I do that? 1844 01:38:14,350 --> 01:38:19,540 And then the whole thing I can say to the 3/2 1845 01:38:19,540 --> 01:38:24,530 or I can use the square root, whichever is your favorite. 1846 01:38:24,530 --> 01:38:28,990 Now, what is k of 0? 1847 01:38:28,990 --> 01:38:29,490 STUDENT: 0. 1848 01:38:29,490 --> 01:38:32,990 Or 1. 1849 01:38:32,990 --> 01:38:34,462 PROFESSOR TODA: Really? 1850 01:38:34,462 --> 01:38:36,226 STUDENT: 1/2. 1851 01:38:36,226 --> 01:38:36,961 3/2. 1852 01:38:36,961 --> 01:38:38,710 PROFESSOR TODA: So let's take this slowly. 1853 01:38:38,710 --> 01:38:43,584 Because we can all make mistakes, goofy mistakes. 1854 01:38:43,584 --> 01:38:45,000 That doesn't mean we're not smart. 1855 01:38:45,000 --> 01:38:46,770 We're very smart, right? 1856 01:38:46,770 --> 01:38:51,210 But it's just a matter of book-keeping and paying 1857 01:38:51,210 --> 01:38:53,010 attention, being attentive. 1858 01:38:53,010 --> 01:38:55,420 OK. 1859 01:38:55,420 --> 01:39:00,240 When I take 0 and replace-- this is drying fast. 1860 01:39:00,240 --> 01:39:02,858 I'm trying to draw it. 1861 01:39:02,858 --> 01:39:10,000 I have 1 over 1 plus 1 to the 3/2. 1862 01:39:10,000 --> 01:39:15,160 I have a student in one exam who was just-- I don't know. 1863 01:39:15,160 --> 01:39:16,580 He was rushing. 1864 01:39:16,580 --> 01:39:20,950 He didn't realize that he had to take it slowly. 1865 01:39:20,950 --> 01:39:22,790 He was extremely smart, though. 1866 01:39:22,790 --> 01:39:29,760 1 over-- you have that 1 plus 1 is 2. 1867 01:39:29,760 --> 01:39:33,710 2 to the 1/2 would be square root of 2 cubed. 1868 01:39:33,710 --> 01:39:35,890 It would be exactly 2 square root of 2. 1869 01:39:35,890 --> 01:39:39,750 And more you can write this as rationalized. 1870 01:39:39,750 --> 01:39:42,050 Now, I have a question for you. 1871 01:39:42,050 --> 01:39:43,010 [INAUDIBLE] 1872 01:39:43,010 --> 01:39:47,099 I'm When we were kids, if you remember-- you are too young. 1873 01:39:47,099 --> 01:39:48,140 Maybe you don't remember. 1874 01:39:48,140 --> 01:39:52,710 But I remember when I was a kid, my teacher would always ask me, 1875 01:39:52,710 --> 01:39:53,980 rationalize your answer. 1876 01:39:53,980 --> 01:39:56,920 Rationalize your answer. 1877 01:39:56,920 --> 01:40:00,362 Put the rational number in the denominator. 1878 01:40:00,362 --> 01:40:02,740 Why do you think that was? 1879 01:40:02,740 --> 01:40:05,156 For hundreds of years people did that. 1880 01:40:05,156 --> 01:40:07,430 STUDENT: [INAUDIBLE] 1881 01:40:07,430 --> 01:40:11,900 PROFESSOR TODA: Because they didn't have a calculator. 1882 01:40:11,900 --> 01:40:16,303 So we used to, even I used to be able to get the square root out 1883 01:40:16,303 --> 01:40:17,518 by hand. 1884 01:40:17,518 --> 01:40:20,920 Has anybody taught you how to compute square root by hand? 1885 01:40:20,920 --> 01:40:21,649 You know that. 1886 01:40:21,649 --> 01:40:22,378 Who taught you? 1887 01:40:22,378 --> 01:40:23,350 STUDENT: I don't remember it. 1888 01:40:23,350 --> 01:40:24,940 My seventh grade teacher taught us. 1889 01:40:24,940 --> 01:40:26,440 PROFESSOR TODA: There is a technique 1890 01:40:26,440 --> 01:40:29,450 of taking groups of twos and then fitting the-- 1891 01:40:29,450 --> 01:40:31,020 and they still teach that. 1892 01:40:31,020 --> 01:40:33,250 I was amazed, they still teach that 1893 01:40:33,250 --> 01:40:35,460 in half of the Asian countries. 1894 01:40:35,460 --> 01:40:39,360 And it's hard, but kids in fifth and sixth grade 1895 01:40:39,360 --> 01:40:45,190 have that practice, which some of us learned and forgot about. 1896 01:40:45,190 --> 01:40:50,029 So imagine that how people would have done this, and of course, 1897 01:40:50,029 --> 01:40:51,070 square root of 2 is easy. 1898 01:40:51,070 --> 01:40:53,550 1.4142, blah blah blah. 1899 01:40:53,550 --> 01:40:54,250 Divide by 2. 1900 01:40:54,250 --> 01:40:56,350 You can do it by hand. 1901 01:40:56,350 --> 01:40:57,860 At least a good approximation. 1902 01:40:57,860 --> 01:41:01,970 But imagine having a nasty square root there to compute, 1903 01:41:01,970 --> 01:41:05,850 and then you would divide by that natural number. 1904 01:41:05,850 --> 01:41:09,140 You have to rely on your own computation to do it. 1905 01:41:09,140 --> 01:41:11,190 There were no calculators. 1906 01:41:11,190 --> 01:41:14,098 How about k of 1? 1907 01:41:14,098 --> 01:41:14,598 How is that? 1908 01:41:14,598 --> 01:41:15,562 What is that? 1909 01:41:15,562 --> 01:41:20,382 1910 01:41:20,382 --> 01:41:23,756 e to the minus 1. 1911 01:41:23,756 --> 01:41:26,540 That's a little bit harder to compute, right? 1912 01:41:26,540 --> 01:41:28,611 1 plus [INAUDIBLE]. 1913 01:41:28,611 --> 01:41:31,760 What is that going to be? 1914 01:41:31,760 --> 01:41:34,040 Minus 2. 1915 01:41:34,040 --> 01:41:37,143 Replace it by 1 to the 3/2. 1916 01:41:37,143 --> 01:41:41,930 I would like you to go home and do the following. 1917 01:41:41,930 --> 01:41:45,810 [INAUDIBLE]-- Not now, not now. 1918 01:41:45,810 --> 01:41:48,310 We stay a little bit longer together. 1919 01:41:48,310 --> 01:41:51,900 k of 0, k of 1/2, and k of 1. 1920 01:41:51,900 --> 01:41:53,034 Which one is bigger? 1921 01:41:53,034 --> 01:41:59,930 1922 01:41:59,930 --> 01:42:03,395 And one last question about that, how much extra credit 1923 01:42:03,395 --> 01:42:04,220 should I give you? 1924 01:42:04,220 --> 01:42:05,870 One point? 1925 01:42:05,870 --> 01:42:08,420 One point if you turn this in. 1926 01:42:08,420 --> 01:42:11,200 Um, yeah. 1927 01:42:11,200 --> 01:42:13,380 Four, [? maybe ?] two points. 1928 01:42:13,380 --> 01:42:19,430 Compare all these three values, and find 1929 01:42:19,430 --> 01:42:32,792 the maximum and the minimum of kappa of t, 1930 01:42:32,792 --> 01:42:37,530 kappa of x, for the interval where 1931 01:42:37,530 --> 01:42:46,690 x is in the interval 0, 1. 1932 01:42:46,690 --> 01:42:48,870 0, closed 1. 1933 01:42:48,870 --> 01:42:49,940 Close it. 1934 01:42:49,940 --> 01:42:53,930 Now, don't ask me, because it's extra credit. 1935 01:42:53,930 --> 01:42:58,740 One question was, by email, can I ask my tutor to help me? 1936 01:42:58,740 --> 01:43:02,170 As long as your tutor doesn't write down your solution, 1937 01:43:02,170 --> 01:43:03,720 you are in good shape. 1938 01:43:03,720 --> 01:43:07,020 Your tutor should help you understand some constants, 1939 01:43:07,020 --> 01:43:08,290 spend time with you. 1940 01:43:08,290 --> 01:43:12,773 But they should not write your assignment themselves. 1941 01:43:12,773 --> 01:43:13,272 OK? 1942 01:43:13,272 --> 01:43:16,540 So it's not a big deal. 1943 01:43:16,540 --> 01:43:22,030 Not I want to tell you one secret that I normally don't 1944 01:43:22,030 --> 01:43:26,730 tell my Calculus 3 students. 1945 01:43:26,730 --> 01:43:29,370 But the more I get to know you, the more 1946 01:43:29,370 --> 01:43:34,020 I realize that you are worth me telling you about that. 1947 01:43:34,020 --> 01:43:35,420 STUDENT: [INAUDIBLE] 1948 01:43:35,420 --> 01:43:38,080 PROFESSOR TODA: No. 1949 01:43:38,080 --> 01:43:41,900 There is a beautiful theory that engineers 1950 01:43:41,900 --> 01:43:48,142 use when they start the motions of curves and parametrizations 1951 01:43:48,142 --> 01:43:51,058 in space. 1952 01:43:51,058 --> 01:43:53,420 And that includes the Frenet formulas. 1953 01:43:53,420 --> 01:43:56,054 1954 01:43:56,054 --> 01:43:58,510 And you already know the first one. 1955 01:43:58,510 --> 01:44:04,520 And I was debating, I was just reviewing what I taught you, 1956 01:44:04,520 --> 01:44:07,321 and I was happy with what I taught you. 1957 01:44:07,321 --> 01:44:10,580 And I said, they know about position vector. 1958 01:44:10,580 --> 01:44:13,100 They know about velocity, acceleration. 1959 01:44:13,100 --> 01:44:15,927 They know how to get back and forth from one another. 1960 01:44:15,927 --> 01:44:16,760 They know our claim. 1961 01:44:16,760 --> 01:44:18,468 They know how to [? reparameterize our ?] 1962 01:44:18,468 --> 01:44:19,940 claims. 1963 01:44:19,940 --> 01:44:25,040 They know the [INAUDIBLE] and B. They know already 1964 01:44:25,040 --> 01:44:27,010 the first Frenet formula. 1965 01:44:27,010 --> 01:44:28,010 They know the curvature. 1966 01:44:28,010 --> 01:44:29,992 What else can I teach them? 1967 01:44:29,992 --> 01:44:34,070 I want to show you-- one of you asked me, 1968 01:44:34,070 --> 01:44:38,280 is this all that we should know? 1969 01:44:38,280 --> 01:44:41,875 This is all that a regular student should know in Calculus 1970 01:44:41,875 --> 01:44:43,746 3, but there is more. 1971 01:44:43,746 --> 01:44:44,870 And you are honor students. 1972 01:44:44,870 --> 01:44:49,930 And I want to show you some beautiful equations here. 1973 01:44:49,930 --> 01:44:54,930 So do you remember that if I introduce r of s 1974 01:44:54,930 --> 01:45:03,900 as a curving arclength, that is a regular curve. 1975 01:45:03,900 --> 01:45:11,050 I said there is a certain famous formula that is T prime of s 1976 01:45:11,050 --> 01:45:13,900 called-- leave space. 1977 01:45:13,900 --> 01:45:15,320 Leave a little bit of space. 1978 01:45:15,320 --> 01:45:15,945 You'll see why. 1979 01:45:15,945 --> 01:45:17,850 It's a surprise. 1980 01:45:17,850 --> 01:45:22,950 k times-- why don't I say k of s? 1981 01:45:22,950 --> 01:45:26,220 Because I want to point out that k is an invariant. 1982 01:45:26,220 --> 01:45:28,580 Even if you have another parameter, 1983 01:45:28,580 --> 01:45:29,905 would be the same function. 1984 01:45:29,905 --> 01:45:38,565 But yes, as a function of s, would be k times N bar, bar. 1985 01:45:38,565 --> 01:45:40,906 More bars because they are free vectors. 1986 01:45:40,906 --> 01:45:42,920 They are not bound to a certain point. 1987 01:45:42,920 --> 01:45:44,550 They're not married to a certain point. 1988 01:45:44,550 --> 01:45:49,180 They are free to shift by parallelism in space. 1989 01:45:49,180 --> 01:45:54,250 However, I'm going to review them as bound at the point 1990 01:45:54,250 --> 01:45:55,200 where they are. 1991 01:45:55,200 --> 01:45:58,110 So they-- no way they are married to the point 1992 01:45:58,110 --> 01:46:03,640 that they belong to. 1993 01:46:03,640 --> 01:46:07,200 Maybe the [? bend ?] will change. 1994 01:46:07,200 --> 01:46:09,394 I don't know how it's going to change like crazy. 1995 01:46:09,394 --> 01:46:18,170 1996 01:46:18,170 --> 01:46:19,250 Something like that. 1997 01:46:19,250 --> 01:46:26,819 At every point you have a T, an N, and it's a 90 degree angle. 1998 01:46:26,819 --> 01:46:30,900 Then you have the binormal, which makes a 90 degree 1999 01:46:30,900 --> 01:46:33,280 angle-- [INAUDIBLE]. 2000 01:46:33,280 --> 01:46:36,860 So the way you should imagine these corners 2001 01:46:36,860 --> 01:46:39,360 would be something like that, right? 2002 01:46:39,360 --> 01:46:40,860 90-90-90. 2003 01:46:40,860 --> 01:46:43,360 It's just hard to draw them. 2004 01:46:43,360 --> 01:46:51,730 Between the vectors you have-- If you draw T and N, am I 2005 01:46:51,730 --> 01:46:53,430 right, that is coming out? 2006 01:46:53,430 --> 01:46:54,110 No. 2007 01:46:54,110 --> 01:46:56,050 I have to switch them. 2008 01:46:56,050 --> 01:46:57,650 T and N. Now, am I right? 2009 01:46:57,650 --> 01:46:59,340 Now I'm thinking of the [? faucet. ?] 2010 01:46:59,340 --> 01:47:02,440 If I move T-- yeah, now it's coming out. 2011 01:47:02,440 --> 01:47:08,120 So this is not getting into the formula. 2012 01:47:08,120 --> 01:47:09,490 So this is the first formula. 2013 01:47:09,490 --> 01:47:10,350 You say, so what? 2014 01:47:10,350 --> 01:47:11,185 You've taught that. 2015 01:47:11,185 --> 01:47:12,450 We proved it together. 2016 01:47:12,450 --> 01:47:14,070 What do you want from us? 2017 01:47:14,070 --> 01:47:17,560 I want to teach you two more formulas. 2018 01:47:17,560 --> 01:47:18,540 N prime. 2019 01:47:18,540 --> 01:47:21,970 2020 01:47:21,970 --> 01:47:24,420 And I'd like you to leave more space here. 2021 01:47:24,420 --> 01:47:27,360 2022 01:47:27,360 --> 01:47:30,746 So you have like an empty field here and an empty field here 2023 01:47:30,746 --> 01:47:32,024 [INAUDIBLE]. 2024 01:47:32,024 --> 01:47:35,815 If you were to compute T prime, the magic thing 2025 01:47:35,815 --> 01:47:40,452 is that T prime is a vector. 2026 01:47:40,452 --> 01:47:41,378 N prime is a vector. 2027 01:47:41,378 --> 01:47:42,770 B prime is a vector. 2028 01:47:42,770 --> 01:47:44,430 They're all vectors. 2029 01:47:44,430 --> 01:47:48,630 They are the derivatives of the vectors T and NB. 2030 01:47:48,630 --> 01:47:50,970 And you say, why would I care about the derivatives 2031 01:47:50,970 --> 01:47:52,210 of the vectors T and NB? 2032 01:47:52,210 --> 01:47:54,310 I'll tell you in a second. 2033 01:47:54,310 --> 01:47:58,070 So if you were to compute in prime, 2034 01:47:58,070 --> 01:48:00,050 you're going to get here. 2035 01:48:00,050 --> 01:48:04,357 Minus k of s times T of s. 2036 01:48:04,357 --> 01:48:07,219 Leave room. 2037 01:48:07,219 --> 01:48:09,570 Leave room, because there is no component that 2038 01:48:09,570 --> 01:48:13,760 depends on N. No such component that that depends on N. 2039 01:48:13,760 --> 01:48:14,680 This is [INAUDIBLE]. 2040 01:48:14,680 --> 01:48:17,260 There is nothing in N. And then in the end 2041 01:48:17,260 --> 01:48:28,580 you'll say, plus tau of s times B. There is missing-- 2042 01:48:28,580 --> 01:48:30,350 something is. 2043 01:48:30,350 --> 01:48:32,990 And finally, if you take B prime, 2044 01:48:32,990 --> 01:48:34,950 there is nothing here, nothing here. 2045 01:48:34,950 --> 01:48:42,865 In the middle you have minus tau of s times N of s. 2046 01:48:42,865 --> 01:48:45,700 2047 01:48:45,700 --> 01:48:49,730 And now you know that nobody else but you knows that. 2048 01:48:49,730 --> 01:48:54,060 The other regular sections don't know these formulas. 2049 01:48:54,060 --> 01:48:57,420 2050 01:48:57,420 --> 01:49:02,220 What do you observe about this bunch of equations? 2051 01:49:02,220 --> 01:49:04,160 Say, oh, wait a minute. 2052 01:49:04,160 --> 01:49:06,520 First of all, why did you put it like that? 2053 01:49:06,520 --> 01:49:07,840 Looks like a cross. 2054 01:49:07,840 --> 01:49:09,160 It is a cross. 2055 01:49:09,160 --> 01:49:12,830 It is like one is shaped in the name of the Father, of the Son, 2056 01:49:12,830 --> 01:49:13,730 and so on. 2057 01:49:13,730 --> 01:49:17,040 So does it have anything to do with religion? 2058 01:49:17,040 --> 01:49:17,540 No. 2059 01:49:17,540 --> 01:49:23,260 But it's going to help you memorize better the equations. 2060 01:49:23,260 --> 01:49:27,004 These are the famous Frenet equations. 2061 01:49:27,004 --> 01:49:30,310 2062 01:49:30,310 --> 01:49:33,980 You only saw the first one. 2063 01:49:33,980 --> 01:49:35,062 What do they represent? 2064 01:49:35,062 --> 01:49:38,230 2065 01:49:38,230 --> 01:49:40,090 If somebody asks you, what is k? 2066 01:49:40,090 --> 01:49:42,910 What it is k of s? 2067 01:49:42,910 --> 01:49:44,130 What's the curvature? 2068 01:49:44,130 --> 01:49:44,880 You go to a party. 2069 01:49:44,880 --> 01:49:46,820 There are only nerds. 2070 01:49:46,820 --> 01:49:47,440 It's you. 2071 01:49:47,440 --> 01:49:50,370 Some people taking advanced calculus or some people 2072 01:49:50,370 --> 01:49:54,790 from Physics, and they say, OK, have you heard of the Frenet 2073 01:49:54,790 --> 01:49:56,920 motion, Frenet formulas, and you say, 2074 01:49:56,920 --> 01:49:58,760 I know everything about it. 2075 01:49:58,760 --> 01:50:02,310 What if they ask you, what is the curvature of k? 2076 01:50:02,310 --> 01:50:07,640 You say, curvature measures how a curve is bent. 2077 01:50:07,640 --> 01:50:11,820 And they say, yeah, but the Frenet formula tells you 2078 01:50:11,820 --> 01:50:13,610 more about that. 2079 01:50:13,610 --> 01:50:17,720 Not only k shows you how bent the curve is. 2080 01:50:17,720 --> 01:50:27,080 But k is a measure of how fast T changes. 2081 01:50:27,080 --> 01:50:28,240 And he sees why. 2082 01:50:28,240 --> 01:50:31,030 Practically, if you take the [INAUDIBLE] to the bat, 2083 01:50:31,030 --> 01:50:37,310 this is the speed of T. So how fast the teaching will change. 2084 01:50:37,310 --> 01:50:39,890 That will be magnitude, will be just k. 2085 01:50:39,890 --> 01:50:42,440 Because magnitude of N is 1. 2086 01:50:42,440 --> 01:50:48,820 So note that k of s is the length of T prime. 2087 01:50:48,820 --> 01:51:04,387 This measures the change in T. So how fast T varies. 2088 01:51:04,387 --> 01:51:08,610 2089 01:51:08,610 --> 01:51:11,320 What does the torsion represent? 2090 01:51:11,320 --> 01:51:16,690 Well, how fast the binormal varies. 2091 01:51:16,690 --> 01:51:20,596 But if you want to think of a helix, 2092 01:51:20,596 --> 01:51:25,640 and it's a little bit hard to imagine, 2093 01:51:25,640 --> 01:51:30,160 the curvature measures how bent a certain curve is. 2094 01:51:30,160 --> 01:51:33,800 And it measures how bent a plane curve is. 2095 01:51:33,800 --> 01:51:38,720 For example, for the circle you have radius a, 1/a, and so on. 2096 01:51:38,720 --> 01:51:40,870 But there must be also a function that 2097 01:51:40,870 --> 01:51:46,090 shows you how a curve twists. 2098 01:51:46,090 --> 01:51:50,060 Because you have not just a plane curve where 2099 01:51:50,060 --> 01:51:52,370 you care about curvature only. 2100 01:51:52,370 --> 01:51:58,570 But in the space curve you care how the curves twist. 2101 01:51:58,570 --> 01:52:03,190 How fast do they move away from a certain plane? 2102 01:52:03,190 --> 01:52:10,956 Now, if I were to draw-- is it hard to memorize these? 2103 01:52:10,956 --> 01:52:11,456 No. 2104 01:52:11,456 --> 01:52:14,060 I memorized them easily based on the fact 2105 01:52:14,060 --> 01:52:19,850 that everything looks like a decomposition 2106 01:52:19,850 --> 01:52:23,920 of a vector in terms of T, N, and B. So in my mind 2107 01:52:23,920 --> 01:52:28,470 it was like, I take any vector I want, B. And this is T, 2108 01:52:28,470 --> 01:52:32,700 this is N, and this is B. Just the weight was IJK. 2109 01:52:32,700 --> 01:52:36,680 Instead if I, I have T. Instead of J, I have N. Instead of K, 2110 01:52:36,680 --> 01:52:40,040 I have B. They are still unit vectors. 2111 01:52:40,040 --> 01:52:42,610 So locally at the point I have this frame 2112 01:52:42,610 --> 01:52:44,230 and I have any vector. 2113 01:52:44,230 --> 01:52:46,950 This vector-- I'm a physicist. 2114 01:52:46,950 --> 01:52:50,940 So let's say I'm going to represent that as v1 times 2115 01:52:50,940 --> 01:52:54,050 the T plus v2 times-- instead of J, 2116 01:52:54,050 --> 01:52:57,790 we'll use that N plus B3 times-- that's 2117 01:52:57,790 --> 01:52:59,980 the last element of the bases. 2118 01:52:59,980 --> 01:53:03,605 Instead of k I have v. So it's the same here. 2119 01:53:03,605 --> 01:53:06,360 You try to pick a vector and decompose 2120 01:53:06,360 --> 01:53:09,950 that in terms of T, N, and B. Will I put that on the final? 2121 01:53:09,950 --> 01:53:10,720 No. 2122 01:53:10,720 --> 01:53:12,925 But I would like you to remember it, especially 2123 01:53:12,925 --> 01:53:17,070 if you are an engineering major or physics major, 2124 01:53:17,070 --> 01:53:19,692 that there is this kind of Frenet frame. 2125 01:53:19,692 --> 01:53:26,010 For those of you who are taking a-- for differential equations, 2126 01:53:26,010 --> 01:53:28,760 you already do some matrices and built-in systems 2127 01:53:28,760 --> 01:53:31,590 of equations, systems of differential equations. 2128 01:53:31,590 --> 01:53:33,215 I'm not going to get there. 2129 01:53:33,215 --> 01:53:38,072 But suppose you don't know differential equations, 2130 01:53:38,072 --> 01:53:41,730 but you know a little bit of linear algebra. 2131 01:53:41,730 --> 01:53:44,950 And I know you know how to multiply matrices. 2132 01:53:44,950 --> 01:53:47,120 You know how I know you multiply matrices, 2133 01:53:47,120 --> 01:53:49,540 no matter how much mathematics you learn. 2134 01:53:49,540 --> 01:53:52,670 And most of you, you are not in general algebra this semester. 2135 01:53:52,670 --> 01:53:55,070 Only two of you are in general algebra. 2136 01:53:55,070 --> 01:54:03,250 When I took a C++ course, the first homework I got was 2137 01:54:03,250 --> 01:54:06,530 to program a matrix multiplication. 2138 01:54:06,530 --> 01:54:07,730 I have to give in matrices. 2139 01:54:07,730 --> 01:54:10,900 I have to program that in C++. 2140 01:54:10,900 --> 01:54:14,600 And freshmen knew that. 2141 01:54:14,600 --> 01:54:20,440 So that means you know how to write this as a matrix 2142 01:54:20,440 --> 01:54:21,410 multiplication. 2143 01:54:21,410 --> 01:54:23,050 Can anybody help me? 2144 01:54:23,050 --> 01:54:25,880 So T, N, B is the magic triple. 2145 01:54:25,880 --> 01:54:28,980 T, N, B's the magic corner. 2146 01:54:28,980 --> 01:54:32,000 T, N, and B are the Three Musketeers who are all 2147 01:54:32,000 --> 01:54:34,326 orthogonal to one another. 2148 01:54:34,326 --> 01:54:37,985 And then I do derivative with respect to s. 2149 01:54:37,985 --> 01:54:42,290 If I want to be elegant, I'll put d/ds. 2150 01:54:42,290 --> 01:54:44,280 OK. 2151 01:54:44,280 --> 01:54:47,330 How am I going to fill in this matrix? 2152 01:54:47,330 --> 01:54:50,650 So somebody who wants to know about differential equations, 2153 01:54:50,650 --> 01:54:51,610 this would be a-- 2154 01:54:51,610 --> 01:54:52,790 STUDENT: 0, k, 0. 2155 01:54:52,790 --> 01:54:53,873 PROFESSOR TODA: Very good. 2156 01:54:53,873 --> 01:55:04,770 0, k, 0, minus k 0 tau, 0 minus tau 0. 2157 01:55:04,770 --> 01:55:07,362 This is called the skew symmetric matrix. 2158 01:55:07,362 --> 01:55:11,810 2159 01:55:11,810 --> 01:55:14,740 Such matrices are very important in robotics. 2160 01:55:14,740 --> 01:55:17,430 If you've ever been to a robotics team, 2161 01:55:17,430 --> 01:55:20,040 like one of those projects, you should 2162 01:55:20,040 --> 01:55:22,990 know that when we study motions of-- let's say 2163 01:55:22,990 --> 01:55:26,620 that my arm performs two rotations in a row. 2164 01:55:26,620 --> 01:55:30,500 All these motions are described based 2165 01:55:30,500 --> 01:55:35,320 on some groups of rotations. 2166 01:55:35,320 --> 01:55:39,950 And if I go into details, it's going to be really hard. 2167 01:55:39,950 --> 01:55:45,580 But practically in such a setting 2168 01:55:45,580 --> 01:55:49,800 we have to deal with matrices that either have determined 2169 01:55:49,800 --> 01:55:53,520 one, like all rotations actually have, 2170 01:55:53,520 --> 01:55:58,300 or have some other properties, like this guy. 2171 01:55:58,300 --> 01:56:00,410 What's the determinant of this guy? 2172 01:56:00,410 --> 01:56:02,010 What do you guys think? 2173 01:56:02,010 --> 01:56:02,750 Just look at it. 2174 01:56:02,750 --> 01:56:03,250 STUDENT: 0? 2175 01:56:03,250 --> 01:56:04,000 PROFESSOR TODA: 0. 2176 01:56:04,000 --> 01:56:05,660 It has determinant 0. 2177 01:56:05,660 --> 01:56:08,470 And moreover, it looks in the mirror. 2178 01:56:08,470 --> 01:56:11,195 So this comes from a group of motion, 2179 01:56:11,195 --> 01:56:14,690 which is little s over 3, the linear algebra, actually. 2180 01:56:14,690 --> 01:56:17,190 So when k is looking in the mirror, 2181 01:56:17,190 --> 01:56:20,820 it becomes minus k tau, is becoming minus tau. 2182 01:56:20,820 --> 01:56:24,190 It is antisymmetric or skew symmetric. 2183 01:56:24,190 --> 01:56:27,010 Skew symmetric or antisymmetric is the same. 2184 01:56:27,010 --> 01:56:29,960 STUDENT: Antisymmetric, skew symmetric matrix. 2185 01:56:29,960 --> 01:56:31,935 PROFESSOR TODA: Skew symmetric or antisymmetric 2186 01:56:31,935 --> 01:56:33,370 is exactly the same thing. 2187 01:56:33,370 --> 01:56:34,292 They are synonyms. 2188 01:56:34,292 --> 01:56:37,060 2189 01:56:37,060 --> 01:56:40,130 So it looks in the mirror and picks up the minus sign, 2190 01:56:40,130 --> 01:56:41,820 has 0 in the bag. 2191 01:56:41,820 --> 01:56:43,135 What am I going to put here? 2192 01:56:43,135 --> 01:56:44,460 You already got the idea. 2193 01:56:44,460 --> 01:56:47,060 So when Ryan gave me this, he meant 2194 01:56:47,060 --> 01:56:50,460 that he knew what I'm going to put here, as a vector, 2195 01:56:50,460 --> 01:56:54,024 as a column vector. 2196 01:56:54,024 --> 01:56:54,936 STUDENT: [INAUDIBLE] 2197 01:56:54,936 --> 01:56:56,019 PROFESSOR TODA: No, no no. 2198 01:56:56,019 --> 01:56:57,040 How do I multiply? 2199 01:56:57,040 --> 01:56:58,510 TNB, right? 2200 01:56:58,510 --> 01:57:01,300 So guys, how do you multiply matrices? 2201 01:57:01,300 --> 01:57:05,420 You go first row and first column. 2202 01:57:05,420 --> 01:57:06,400 So you go like this. 2203 01:57:06,400 --> 01:57:13,629 0 times T plus k times 10 plus 0 times B. Here it is. 2204 01:57:13,629 --> 01:57:15,170 So I'm teaching you a little bit more 2205 01:57:15,170 --> 01:57:18,620 than-- if you are going to stick with linear algebra 2206 01:57:18,620 --> 01:57:21,250 and stick with differential equations, 2207 01:57:21,250 --> 01:57:25,440 this is a good introduction to more of those mathematics. 2208 01:57:25,440 --> 01:57:26,111 Yes, sir? 2209 01:57:26,111 --> 01:57:28,059 STUDENT: Why don't you use Cramer's rule? 2210 01:57:28,059 --> 01:57:28,850 PROFESSOR TODA: Uh? 2211 01:57:28,850 --> 01:57:31,270 STUDENT: Why don't you use the Cramer's rule? 2212 01:57:31,270 --> 01:57:32,722 PROFESSOR TODA: The Cramer's rule? 2213 01:57:32,722 --> 01:57:34,835 STUDENT: Yeah. [INAUDIBLE]. 2214 01:57:34,835 --> 01:57:35,626 PROFESSOR TODA: No. 2215 01:57:35,626 --> 01:57:44,070 First of all, Crarmer's rule is to solve systems of equations 2216 01:57:44,070 --> 01:57:47,810 that don't involve derivatives, like a linear system 2217 01:57:47,810 --> 01:57:51,960 like Ax equals B. I'm going to have, 2218 01:57:51,960 --> 01:57:56,760 for example, 3x1 plus 2x3 equals 1. 2219 01:57:56,760 --> 01:58:01,000 5x1 plus x2 plus x3 equals something else. 2220 01:58:01,000 --> 01:58:03,400 So for that I can use Cramer's rule. 2221 01:58:03,400 --> 01:58:04,690 But look at that! 2222 01:58:04,690 --> 01:58:06,060 This is really complicated. 2223 01:58:06,060 --> 01:58:07,840 It's a dynamical system. 2224 01:58:07,840 --> 01:58:11,580 At every moment of time the vectors are changing. 2225 01:58:11,580 --> 01:58:13,420 So it's a crazy [INAUDIBLE]. 2226 01:58:13,420 --> 01:58:19,095 Like A of t times something, so some vector 2227 01:58:19,095 --> 01:58:22,858 that is also depending on time equals the derivative 2228 01:58:22,858 --> 01:58:25,000 of that vector that [INAUDIBLE]. 2229 01:58:25,000 --> 01:58:31,560 So that's a OD system that you should learn in 3351. 2230 01:58:31,560 --> 01:58:33,360 So I don't know what your degree plan is, 2231 01:58:33,360 --> 01:58:35,162 but most of you in engineering will 2232 01:58:35,162 --> 01:58:43,802 take my class, 2316 in algebra, OD1 3350 where they teach you 2233 01:58:43,802 --> 01:58:45,010 about differential equations. 2234 01:58:45,010 --> 01:58:48,070 These are all differential equations, all three of them. 2235 01:58:48,070 --> 01:58:51,326 In 3351 you learn about this system 2236 01:58:51,326 --> 01:58:54,460 which is a system of differential equation. 2237 01:58:54,460 --> 01:58:57,210 And then you practically say, now I 2238 01:58:57,210 --> 01:58:59,840 know everything I need to know in math, and you say, 2239 01:58:59,840 --> 01:59:01,100 goodbye math. 2240 01:59:01,100 --> 01:59:02,740 If you guys wanted to learn more, 2241 01:59:02,740 --> 01:59:06,223 of course I would be very happy to learn that, hey, I 2242 01:59:06,223 --> 01:59:08,810 like math, I'd like to be a double major. 2243 01:59:08,810 --> 01:59:12,320 I'd like to be not just an engineering, but also math 2244 01:59:12,320 --> 01:59:14,550 major if you really like it. 2245 01:59:14,550 --> 01:59:18,170 Many people already have a minor. 2246 01:59:18,170 --> 01:59:20,240 Many of you have a minor in your plan. 2247 01:59:20,240 --> 01:59:22,835 Like for that minor you only need-- 2248 01:59:22,835 --> 01:59:24,170 STUDENT: One extra math course. 2249 01:59:24,170 --> 01:59:25,753 PROFESSOR TODA: One extra math course. 2250 01:59:25,753 --> 01:59:30,720 For example, with 3350 you don't need 3351 for a minor. 2251 01:59:30,720 --> 01:59:31,380 Why? 2252 01:59:31,380 --> 01:59:34,380 Because you are taking the probability in stats anyway. 2253 01:59:34,380 --> 01:59:35,260 You have to. 2254 01:59:35,260 --> 01:59:38,600 They force you to do that, 3342. 2255 01:59:38,600 --> 01:59:44,970 So if you take 3351 it's on top of the minor that we give you. 2256 01:59:44,970 --> 01:59:46,400 I know because that's what I do. 2257 01:59:46,400 --> 01:59:47,980 I look at the degree plans. 2258 01:59:47,980 --> 01:59:51,960 And I work closely to the math adviser, with Patty. 2259 01:59:51,960 --> 01:59:54,265 She has all the [INAUDIBLE]. 2260 01:59:54,265 --> 01:59:55,390 STUDENT: So is [INAUDIBLE]? 2261 01:59:55,390 --> 01:59:59,527 2262 01:59:59,527 --> 02:00:00,860 PROFESSOR TODA: You mean double? 2263 02:00:00,860 --> 02:00:01,840 Double degree? 2264 02:00:01,840 --> 02:00:04,214 We have this already in place. 2265 02:00:04,214 --> 02:00:05,380 We've had it for many years. 2266 02:00:05,380 --> 02:00:07,400 It's an excellent plan. 2267 02:00:07,400 --> 02:00:09,870 162 hours it is now. 2268 02:00:09,870 --> 02:00:12,870 It used to be 159. 2269 02:00:12,870 --> 02:00:17,700 Double major, computer science and mathematics. 2270 02:00:17,700 --> 02:00:22,840 And I could say they were some of the most successful 2271 02:00:22,840 --> 02:00:26,520 in terms of finding jobs. 2272 02:00:26,520 --> 02:00:28,660 What would you take on top of that? 2273 02:00:28,660 --> 02:00:30,945 Well, as a math major you have a few more courses 2274 02:00:30,945 --> 02:00:32,860 to take one top of that. 2275 02:00:32,860 --> 02:00:36,720 You can link your computer science with the mathematics, 2276 02:00:36,720 --> 02:00:39,630 for example, by taking numerical analysis. 2277 02:00:39,630 --> 02:00:42,300 If you love computers and you like calculus 2278 02:00:42,300 --> 02:00:46,700 and you want to put together all the information 2279 02:00:46,700 --> 02:00:49,140 you have in both, then numerical analysis 2280 02:00:49,140 --> 02:00:50,430 would be your best bet. 2281 02:00:50,430 --> 02:00:55,215 And they require that in both computer science degree 2282 02:00:55,215 --> 02:00:58,230 if you are a double major, and your math degree. 2283 02:00:58,230 --> 02:01:03,050 So the good thing is that some things count for both degrees. 2284 02:01:03,050 --> 02:01:06,746 And so with those 160 hours you are very happy. 2285 02:01:06,746 --> 02:01:10,060 Oh, I'm done, I got a few more hours. 2286 02:01:10,060 --> 02:01:12,420 Many math majors already have around 130. 2287 02:01:12,420 --> 02:01:13,830 They're not supposed to. 2288 02:01:13,830 --> 02:01:16,080 They are supposed to stop at 120. 2289 02:01:16,080 --> 02:01:19,690 So why not go the extra 20 hours and get two degrees in one? 2290 02:01:19,690 --> 02:01:21,133 STUDENT: It's a semester. 2291 02:01:21,133 --> 02:01:22,095 PROFESSOR TODA: Yeah. 2292 02:01:22,095 --> 02:01:23,428 Of course, it's a lot more work. 2293 02:01:23,428 --> 02:01:26,430 But we have people who like-- really they 2294 02:01:26,430 --> 02:01:30,170 are nerdy people who loved computer science from when 2295 02:01:30,170 --> 02:01:31,990 they were three or four. 2296 02:01:31,990 --> 02:01:33,410 And they also like math. 2297 02:01:33,410 --> 02:01:37,160 And they say, OK, I want to do both. 2298 02:01:37,160 --> 02:01:41,640 OK, a little bit more and I'll let you go. 2299 02:01:41,640 --> 02:01:44,822 Now I want you to ask me other questions 2300 02:01:44,822 --> 02:01:48,486 you may have had about the homework, anything that 2301 02:01:48,486 --> 02:01:59,224 gave you headache, anything that you feel you need a little bit 2302 02:01:59,224 --> 02:02:00,712 more of an explanation about. 2303 02:02:00,712 --> 02:02:12,471 2304 02:02:12,471 --> 02:02:12,970 Yes? 2305 02:02:12,970 --> 02:02:14,011 STUDENT: I just have one. 2306 02:02:14,011 --> 02:02:15,795 In WeBWork, what is the easiest way 2307 02:02:15,795 --> 02:02:17,650 to take the square root of something? 2308 02:02:17,650 --> 02:02:18,610 STUDENT: sqrt. 2309 02:02:18,610 --> 02:02:21,966 PROFESSOR TODA: sqrt is what you type. 2310 02:02:21,966 --> 02:02:24,870 But of course you can also go to the caret 1/2. 2311 02:02:24,870 --> 02:02:27,930 2312 02:02:27,930 --> 02:02:29,490 Something non-technical? 2313 02:02:29,490 --> 02:02:34,354 Any question, yes sir, from the homework? 2314 02:02:34,354 --> 02:02:38,690 Or in relation to [INAUDIBLE]? 2315 02:02:38,690 --> 02:02:41,506 STUDENT: I don't understand why is the tangent unit vector, 2316 02:02:41,506 --> 02:02:44,162 it's just the slope off of that line, right? 2317 02:02:44,162 --> 02:02:45,144 The drunk bug? 2318 02:02:45,144 --> 02:02:47,100 Whatever line the drunk bug is on? 2319 02:02:47,100 --> 02:02:49,120 PROFESSOR TODA: So it would be the tangent 2320 02:02:49,120 --> 02:02:52,040 to the directional motion, which is a curve. 2321 02:02:52,040 --> 02:02:54,620 2322 02:02:54,620 --> 02:02:58,140 And normalized to have length one. 2323 02:02:58,140 --> 02:03:01,700 Because otherwise our prime is-- you may say, 2324 02:03:01,700 --> 02:03:04,210 why do you need T to be unitary? 2325 02:03:04,210 --> 02:03:07,150 2326 02:03:07,150 --> 02:03:11,355 OK, computations become horrible unless your speed 2327 02:03:11,355 --> 02:03:13,815 is 1 or 5 or 9. 2328 02:03:13,815 --> 02:03:18,140 If the speed is a constant, everything else becomes easier. 2329 02:03:18,140 --> 02:03:20,182 So that's one reason. 2330 02:03:20,182 --> 02:03:22,086 STUDENT: And why is the derivative 2331 02:03:22,086 --> 02:03:24,466 of T then perpendicular? 2332 02:03:24,466 --> 02:03:26,502 Why does it always turn into-- 2333 02:03:26,502 --> 02:03:27,960 PROFESSOR TODA: Perpendicular to T? 2334 02:03:27,960 --> 02:03:30,940 We've done that last time, but I'm glad to do it again. 2335 02:03:30,940 --> 02:03:34,430 And I forgot what we wrote in the book, 2336 02:03:34,430 --> 02:03:36,720 and I also saw in the book this thing 2337 02:03:36,720 --> 02:03:42,800 that if you have R, in absolute value, constant-- 2338 02:03:42,800 --> 02:03:44,810 and I've done that with you guys-- 2339 02:03:44,810 --> 02:03:52,020 prove that R and R prime had every point perpendicular. 2340 02:03:52,020 --> 02:03:54,850 So if you have-- we've done that before. 2341 02:03:54,850 --> 02:03:57,090 Now, what do you do then? 2342 02:03:57,090 --> 02:04:00,564 T [INAUDIBLE] T is 1. 2343 02:04:00,564 --> 02:04:04,500 The scalar [INAUDIBLE] the product. 2344 02:04:04,500 --> 02:04:09,540 T prime times T plus T prime T prime. 2345 02:04:09,540 --> 02:04:12,090 So 0. 2346 02:04:12,090 --> 02:04:16,540 And T is perpendicular to T prime, 2347 02:04:16,540 --> 02:04:20,610 because that means T or T prime equals 0. 2348 02:04:20,610 --> 02:04:27,860 2349 02:04:27,860 --> 02:04:30,360 When you run in a circle, you say-- 2350 02:04:30,360 --> 02:04:33,790 OK, let's run in a circle. 2351 02:04:33,790 --> 02:04:40,650 I say, this is my T. I can feel that there is something that's 2352 02:04:40,650 --> 02:04:42,530 trying to bend me this way. 2353 02:04:42,530 --> 02:04:44,366 That is my acceleration. 2354 02:04:44,366 --> 02:04:49,040 And I have to-- but I don't know-- how familiar are you 2355 02:04:49,040 --> 02:04:51,385 with the winter sports? 2356 02:04:51,385 --> 02:04:54,150 2357 02:04:54,150 --> 02:04:58,070 In many winter sports, the Frenet Trihedron is crucial. 2358 02:04:58,070 --> 02:05:01,090 Imagine that you have one of those slopes, 2359 02:05:01,090 --> 02:05:04,890 and all of the sudden the torsion becomes too weak. 2360 02:05:04,890 --> 02:05:06,680 That means it becomes dangerous. 2361 02:05:06,680 --> 02:05:09,870 That means that the vehicle you're in, 2362 02:05:09,870 --> 02:05:15,010 the snow vehicle or any kind of-- your skis, [INAUDIBLE], 2363 02:05:15,010 --> 02:05:20,850 if the torsion of your body moving can become too big, 2364 02:05:20,850 --> 02:05:21,880 that will be a problem. 2365 02:05:21,880 --> 02:05:24,670 So you have to redesign that some more. 2366 02:05:24,670 --> 02:05:26,660 And this is what they do. 2367 02:05:26,660 --> 02:05:28,570 You know there have been many accidents. 2368 02:05:28,570 --> 02:05:32,360 And many times they say, even in Formula One, 2369 02:05:32,360 --> 02:05:38,170 the people who project a certain racetrack, 2370 02:05:38,170 --> 02:05:41,620 like a track in Indianapolis or Montecarlo 2371 02:05:41,620 --> 02:05:44,460 or whatever, they have to have in mind 2372 02:05:44,460 --> 02:05:47,660 that Frenet frame every second. 2373 02:05:47,660 --> 02:05:50,690 So there are simulators showing how 2374 02:05:50,690 --> 02:05:52,874 the Frenet frame is changing. 2375 02:05:52,874 --> 02:05:55,718 There are programs that measure the curvature 2376 02:05:55,718 --> 02:05:59,950 in a torsion for those simulators at every point. 2377 02:05:59,950 --> 02:06:02,690 Neither the curvature nor the torsion 2378 02:06:02,690 --> 02:06:04,360 can exceed a certain value. 2379 02:06:04,360 --> 02:06:06,900 Otherwise it becomes dangerous. 2380 02:06:06,900 --> 02:06:09,545 You say, oh, I thought only the speed is a danger. 2381 02:06:09,545 --> 02:06:10,910 Nope. 2382 02:06:10,910 --> 02:06:14,846 It's also the way that the motion, if it's a skew curve, 2383 02:06:14,846 --> 02:06:16,710 it's really complicated. 2384 02:06:16,710 --> 02:06:20,030 Because you twist and turn and bend in many ways. 2385 02:06:20,030 --> 02:06:22,239 And it can become really dangerous. 2386 02:06:22,239 --> 02:06:23,280 Speed is not [INAUDIBLE]. 2387 02:06:23,280 --> 02:06:26,262 2388 02:06:26,262 --> 02:06:30,252 STUDENT: So the torsion was the twists in the track? 2389 02:06:30,252 --> 02:06:31,960 PROFESSOR TODA: The torsion is the twist. 2390 02:06:31,960 --> 02:06:34,580 And by the way, keep your idea. 2391 02:06:34,580 --> 02:06:37,290 You wanted to ask something more? 2392 02:06:37,290 --> 02:06:43,090 When you twist-- suppose you have something like a race car. 2393 02:06:43,090 --> 02:06:47,190 And the race car is at the walls of the track. 2394 02:06:47,190 --> 02:06:57,980 And here's-- when you have a very abrupt curvature 2395 02:06:57,980 --> 02:07:03,760 and torsion, and you can have that in Formula One as well, 2396 02:07:03,760 --> 02:07:09,922 why do they build one wall a lot higher than the other? 2397 02:07:09,922 --> 02:07:13,980 Because the poor car-- I don't know how passionate you 2398 02:07:13,980 --> 02:07:19,552 are about Formula One or car races-- 2399 02:07:19,552 --> 02:07:24,690 the poor car is going to be close to the wall. 2400 02:07:24,690 --> 02:07:28,382 It's going to bend like that, that wall would be round. 2401 02:07:28,382 --> 02:07:32,850 And as a builder, you have to build the wall really high. 2402 02:07:32,850 --> 02:07:35,818 Because that kind of high speed, high velocity, 2403 02:07:35,818 --> 02:07:39,350 high curvature, the poor car's going szhhhhh-- then 2404 02:07:39,350 --> 02:07:42,050 again on a normal track. 2405 02:07:42,050 --> 02:07:45,140 Imagine what happens if the wall is not high enough. 2406 02:07:45,140 --> 02:07:48,490 The wheels of the car will go up and get over. 2407 02:07:48,490 --> 02:07:50,041 And it's going to be a disaster. 2408 02:07:50,041 --> 02:07:52,740 2409 02:07:52,740 --> 02:07:57,490 So that engineer ha to study all the parametric equations 2410 02:07:57,490 --> 02:08:01,240 and the Frenet frame and deep down make a simulator, 2411 02:08:01,240 --> 02:08:04,450 compute how tall the walls should be in order for the car 2412 02:08:04,450 --> 02:08:10,220 not to get over on the other side or get off the track. 2413 02:08:10,220 --> 02:08:12,350 It's really complicated stuff. 2414 02:08:12,350 --> 02:08:14,890 It's all mathematics and physics, 2415 02:08:14,890 --> 02:08:18,680 but all the applications are run by engineers and-- yes, sir? 2416 02:08:18,680 --> 02:08:22,033 STUDENT: What's the difference [INAUDIBLE] centrifugal force? 2417 02:08:22,033 --> 02:08:23,950 PROFESSOR TODA: The centrifugal force 2418 02:08:23,950 --> 02:08:26,380 is related to our double prime. 2419 02:08:26,380 --> 02:08:32,130 Our double prime is related to N and T at the same time. 2420 02:08:32,130 --> 02:08:36,160 So at some point, let me ask you one last question and I'm done. 2421 02:08:36,160 --> 02:08:39,210 2422 02:08:39,210 --> 02:08:43,236 What's the relationship between acceleration or double prime? 2423 02:08:43,236 --> 02:08:45,982 And are they the same thing? 2424 02:08:45,982 --> 02:08:50,297 And when are they not the same thing? 2425 02:08:50,297 --> 02:08:52,380 Because you say, OK, practically the centrifugal-- 2426 02:08:52,380 --> 02:08:54,180 STUDENT: They're the same on a curve. 2427 02:08:54,180 --> 02:08:55,740 PROFESSOR TODA: They are the same-- 2428 02:08:55,740 --> 02:08:56,823 STUDENT: Like on a circle. 2429 02:08:56,823 --> 02:08:58,520 PROFESSOR TODA: On a circle! 2430 02:08:58,520 --> 02:09:00,090 And you are getting so close. 2431 02:09:00,090 --> 02:09:01,370 It's hot, hot, hot. 2432 02:09:01,370 --> 02:09:08,100 On a circle and on a helix they are the same up to a constant. 2433 02:09:08,100 --> 02:09:11,310 So what do you think the magic answer will be? 2434 02:09:11,310 --> 02:09:12,480 N was what, guys? 2435 02:09:12,480 --> 02:09:15,220 N was-- remind me again. 2436 02:09:15,220 --> 02:09:18,475 That was T prime over absolute value of T prime. 2437 02:09:18,475 --> 02:09:22,290 But that doesn't mean, does not equal, in general, 2438 02:09:22,290 --> 02:09:26,350 does not equal to R double prime. 2439 02:09:26,350 --> 02:09:28,096 When is it equal? 2440 02:09:28,096 --> 02:09:29,930 In general it's not equal. 2441 02:09:29,930 --> 02:09:31,070 When is it equal? 2442 02:09:31,070 --> 02:09:35,440 If you are in aclength, you see the advantage of aclength. 2443 02:09:35,440 --> 02:09:36,965 It's wonderful. 2444 02:09:36,965 --> 02:09:40,067 In arclength, T is R prime of s. 2445 02:09:40,067 --> 02:09:45,940 And in arclength that means T prime is R double prime of s. 2446 02:09:45,940 --> 02:09:48,590 And in arclength I just told you, 2447 02:09:48,590 --> 02:09:50,330 T prime is the first Frenet formula. 2448 02:09:50,330 --> 02:09:55,510 It'll be curvature times the N. 2449 02:09:55,510 --> 02:10:02,180 So the acceleration practically and the N 2450 02:10:02,180 --> 02:10:06,560 will be the same in arclength, up to a scalar multiplication. 2451 02:10:06,560 --> 02:10:11,678 But what if your speed is not even constant? 2452 02:10:11,678 --> 02:10:12,530 Then God help you. 2453 02:10:12,530 --> 02:10:16,850 Because the acceleration R double prime and N 2454 02:10:16,850 --> 02:10:19,780 are not colinear. 2455 02:10:19,780 --> 02:10:24,370 So if I were to draw-- and that's my last picture-- 2456 02:10:24,370 --> 02:10:26,950 let me give you a wild motion here. 2457 02:10:26,950 --> 02:10:32,466 You start slow and then you go crazy and fast and slow down. 2458 02:10:32,466 --> 02:10:35,657 Just like most of the physical models from the bugs 2459 02:10:35,657 --> 02:10:38,900 and the flies and so on. 2460 02:10:38,900 --> 02:10:44,950 In that kind of crazy motion you have a T and N at every point. 2461 02:10:44,950 --> 02:10:45,450 [INAUDIBLE] 2462 02:10:45,450 --> 02:10:48,430 2463 02:10:48,430 --> 02:10:50,860 [? v ?] will be down. 2464 02:10:50,860 --> 02:10:53,360 And T is here. 2465 02:10:53,360 --> 02:10:56,940 So can you draw arc double prime for me? 2466 02:10:56,940 --> 02:10:59,433 It will still be towards the inside. 2467 02:10:59,433 --> 02:11:04,200 But it's still going to coincide with N. Maybe this one. 2468 02:11:04,200 --> 02:11:12,640 What's the magic thing is that T, N, and R double prime 2469 02:11:12,640 --> 02:11:15,745 are in the same plane always. 2470 02:11:15,745 --> 02:11:18,170 That's another secret other students 2471 02:11:18,170 --> 02:11:19,630 don't know in Calculus 3. 2472 02:11:19,630 --> 02:11:22,456 That same thing is called osculating plane. 2473 02:11:22,456 --> 02:11:25,630 2474 02:11:25,630 --> 02:11:31,270 We have a few magic names for these things. 2475 02:11:31,270 --> 02:11:36,509 So T and N, the plane that is-- how shall I say that? 2476 02:11:36,509 --> 02:11:37,050 I don't know. 2477 02:11:37,050 --> 02:11:43,638 The plane given by T and N is called osculating plane. 2478 02:11:43,638 --> 02:11:46,710 2479 02:11:46,710 --> 02:11:49,080 The acceleration is always on that plane. 2480 02:11:49,080 --> 02:11:52,460 So imagine T and N are in the same shaded plane. 2481 02:11:52,460 --> 02:11:55,500 R double prime is in the same plane. 2482 02:11:55,500 --> 02:11:56,550 OK? 2483 02:11:56,550 --> 02:11:58,510 Now, can you guess the other two names? 2484 02:11:58,510 --> 02:12:03,639 So this is T, this is N. And B is up. 2485 02:12:03,639 --> 02:12:04,805 This is my body's direction. 2486 02:12:04,805 --> 02:12:06,200 T and N, look at me. 2487 02:12:06,200 --> 02:12:10,160 T, N, and B. I'm the Frenet Trihedron. 2488 02:12:10,160 --> 02:12:13,360 Which one is the osculating plane? 2489 02:12:13,360 --> 02:12:16,740 It's the horizontal xy plane. 2490 02:12:16,740 --> 02:12:20,940 OK, do you know-- maybe you're a mechanical engineering major, 2491 02:12:20,940 --> 02:12:23,650 and after that I will let you go. 2492 02:12:23,650 --> 02:12:25,920 No extra credit, though for this task. 2493 02:12:25,920 --> 02:12:29,130 Maybe I'm going to start asking questions and give you $1. 2494 02:12:29,130 --> 02:12:31,330 I used to do that a lot in differential equations, 2495 02:12:31,330 --> 02:12:34,580 like ask a hard question, whoever gets it first, 2496 02:12:34,580 --> 02:12:36,420 give her a dollar. 2497 02:12:36,420 --> 02:12:41,595 Until a point when they asked me to teach Honors 3350 when 2498 02:12:41,595 --> 02:12:44,210 I started having three or four people answering the question 2499 02:12:44,210 --> 02:12:44,925 at the same time. 2500 02:12:44,925 --> 02:12:49,020 And that was a significant expense, 2501 02:12:49,020 --> 02:12:52,215 because I had to give $4 away at the same time. 2502 02:12:52,215 --> 02:12:53,840 STUDENT: I feel like you should've just 2503 02:12:53,840 --> 02:12:54,590 split it between-- 2504 02:12:54,590 --> 02:12:57,950 PROFESSOR TODA: So that's normal and binormal. 2505 02:12:57,950 --> 02:13:00,800 This is me, the binormal, and this is the normal. 2506 02:13:00,800 --> 02:13:03,080 Does anybody know the name of this plane, 2507 02:13:03,080 --> 02:13:05,850 between normal and bionormal? 2508 02:13:05,850 --> 02:13:08,170 This would be this plane. 2509 02:13:08,170 --> 02:13:10,750 STUDENT: The skew [INAUDIBLE]. 2510 02:13:10,750 --> 02:13:12,250 PROFESSOR TODA: Normal and binormal. 2511 02:13:12,250 --> 02:13:13,884 They call that normal plane. 2512 02:13:13,884 --> 02:13:16,540 2513 02:13:16,540 --> 02:13:22,510 So it's tricky if you are not a mechanical engineering major. 2514 02:13:22,510 --> 02:13:28,460 But some of you are maybe and will learn that later. 2515 02:13:28,460 --> 02:13:29,940 Any other questions for me? 2516 02:13:29,940 --> 02:13:33,890 Now, in my office I'm going to do review. 2517 02:13:33,890 --> 02:13:37,836 I was wondering if you have time, 2518 02:13:37,836 --> 02:13:39,960 I don't know if you have time to come to my office, 2519 02:13:39,960 --> 02:13:43,340 but should you have any kind of homework related question, 2520 02:13:43,340 --> 02:13:46,290 I'll be very happy to answer it now. 2521 02:13:46,290 --> 02:13:49,000 3:00 to 5:00. 2522 02:13:49,000 --> 02:13:51,020 Now, one time I had a student who 2523 02:13:51,020 --> 02:13:53,175 only had seven questions left. 2524 02:13:53,175 --> 02:13:55,690 He came to my office and he left with no homework. 2525 02:13:55,690 --> 02:13:57,593 We finished all of them. 2526 02:13:57,593 --> 02:13:58,380 And I felt guilty. 2527 02:13:58,380 --> 02:14:00,870 But at the same, he said, well, no, it's 2528 02:14:00,870 --> 02:14:03,170 better I came to you instead of going to my tutor. 2529 02:14:03,170 --> 02:14:05,180 It was fine. 2530 02:14:05,180 --> 02:14:08,670 So we can try some problems together today 2531 02:14:08,670 --> 02:14:11,850 if you want between 3:00 and 5:00, if you have the time. 2532 02:14:11,850 --> 02:14:13,838 Some of you don't have the time. 2533 02:14:13,838 --> 02:14:14,832 All right? 2534 02:14:14,832 --> 02:14:16,323 If you don't have the time today, 2535 02:14:16,323 --> 02:14:19,305 and you would like to be helped [INAUDIBLE], 2536 02:14:19,305 --> 02:14:21,293 click Email Instructor. 2537 02:14:21,293 --> 02:14:24,275 I'm going to get the questions [INAUDIBLE]. 2538 02:14:24,275 --> 02:14:26,014 You're welcome to ask me anything 2539 02:14:26,014 --> 02:14:27,257 at any time over there. 2540 02:14:27,257 --> 02:14:38,191 2541 02:14:38,191 --> 02:14:41,173 [CLASSROOM CHATTER] 2542 02:14:41,173 --> 02:15:12,348 2543 02:15:12,348 --> 02:15:14,472 PROFESSOR TODA: I have somebody who's taking notes. 2544 02:15:14,472 --> 02:15:15,167 STUDENT: Yeah, I know. 2545 02:15:15,167 --> 02:15:15,963 And that's why I was like-- 2546 02:15:15,963 --> 02:15:16,957 PROFESSOR TODA: He's going to make a copy 2547 02:15:16,957 --> 02:15:18,448 and I'll give you a copy. 2548 02:15:18,448 --> 02:15:19,442 STUDENT: Yeah. 2549 02:15:19,442 --> 02:15:23,621 My Cal 1 teacher, Dr. [INAUDIBLE]. 2550 02:15:23,621 --> 02:15:24,412 STUDENT: Thank you. 2551 02:15:24,412 --> 02:15:25,495 PROFESSOR TODA: Yes, yeah. 2552 02:15:25,495 --> 02:15:26,897 Have a nice day. 2553 02:15:26,897 --> 02:15:29,030 STUDENT: --got really mad when I don't take notes. 2554 02:15:29,030 --> 02:15:34,680 Because he felt like I was not, I guess-- 2555 02:15:34,680 --> 02:15:36,503